#Algebra 2. Quadratic Equations/Complete The Square (i think)

422 messages · Page 1 of 1 (latest)

vale moth
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I just joined like an hour ago and got too scared too ask for help but here i am. I have a bunch of questions similar to these because im doing the topic called ''Maximum and Minimum of Quadratic Graphs''. Im pretty sure thats what this question is based on but the problem is, is that i genuinely have absolutely no idea on how i am meant to find anything. Idk if u need to use a formula or find something first or what f(x) even means? If someone could please explain to me or just give me a summary of this WHOLE topic bc i dont understand anything, It would be greatly appreciated

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shut copper
vale moth
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i would like the simplifided explanation first

shut copper
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Okay, when we talk about f(x), we mean x to be a number and f(x) to be the number we get when we plug in the value of x into the expression that f(x) is equal to.

vale moth
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okay

shut copper
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So the "smallest possible value of f(x)" would be talking about, like, if we plugged in all the numbers into x, what would be the smallest f(x) we would get?"

vale moth
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by all do u mean every single number?

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like 1 to inf?

shut copper
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Like -inf to inf.

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Except excluding the infinities, obviously, because they're not numbers.

vale moth
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okay so

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smallest possible value of f(x) is saying that if we put in any number into x, what would be the smallest number we get when we put in the value of x into the expression that f(x) is equal to?

shut copper
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Kind of.

vale moth
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im guessing this involves graphs...

shut copper
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A whole lot of math depends on being very specific.

shut copper
vale moth
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hm

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should i visualise a graph in my head for this?

shut copper
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Maybe?

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It's not necessary.

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I don't know you well enough to know whether it would help or distract.

vale moth
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hmmm

shut copper
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And I'm not sure how you'd be able to visualize the graph well enough without doing the algebraic work that would give you the answer anyway.

vale moth
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okay

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im guessing that some part of this includes completing the square though?

shut copper
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I mean, that's definitely one approach.

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Do you know how completing the square works?

vale moth
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yes

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well

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i see maths as letters and numbers

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im told ur not meant to

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but i do

shut copper
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What do you mean, you're "not meant to"?

vale moth
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so i just follow the b formula thing

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b = smt, b/2 = smt, (b/2)squared is it

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then +_ the value

vale moth
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and not just numbers and letters

shut copper
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I think I might get it. You're memorizing formulae, but you're not understanding the proofs.

vale moth
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yes

shut copper
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So here's how it works.

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I'm gonna show you how this completing the square stuff works on an arbitrary quadratic.

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That is, we're going to start with ax^2 + bx + c, and then I'm gonna show you each step.

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Actually, we should probably start in reverse a bit.

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Do you know how to expand a binomial?

vale moth
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maybe?

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i dont speak maths language

shut copper
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(x + a)^2 equals what?

vale moth
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oh

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x squared plus 2x plus a squared

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i think smt like that

shut copper
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x^2 + 2ax + a^2.

vale moth
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oh right

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sorry

shut copper
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I'll show you the proof.

vale moth
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yes i know that

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i just dont remember any of the maths words like binomial

shut copper
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(x + a)^2 = (x + a)(x + a)
          = x(x + a) + a(x + a)
          = xx + xa + ax + aa
          = x^2 + 2ax + a^2```
vale moth
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yes that is how i do it

shut copper
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So you did know the proof?

vale moth
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the proof as in what?

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OH

shut copper
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...that's the proof that (x + a)^2 = x^2 + 2ax + a^2.

vale moth
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yes.

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but thats mostly because its an easier algebra

shut copper
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What do you mean "an easier algebra"?

vale moth
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i use pascals traingle or the formula for those

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its easier to do

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for me

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i dont know if every country uses a system like algebra 1,2,3

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if they do then i am able to do all of algebra 1

shut copper
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What's important to understand here is that math is a system of logic. If you can't prove a statement true, you can't say that it's true.

vale moth
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yes

shut copper
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So we've proved that (x + a)^2 = x^2 + 2ax + a^2, therefore we are now allowed to use that fact.

vale moth
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okay

shut copper
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So now let's think about ax^2 + bx + c.

vale moth
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okay

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it involes -b formula

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or the 2 brackets

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what multiply by what for c adds up to b

shut copper
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No, let's prove what we're talking about.

vale moth
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okay

shut copper
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Now, we're talking about "completing the square", right?

vale moth
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yes

shut copper
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And it's called that because the goal is to get to something that looks like (x + a)^2.

vale moth
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yes you are correct

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my answers do look like that

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for others

shut copper
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So, we have ax^2 + bx + c.

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So first, we note that (x + a)^2 = x^2 + 2ax + a^2 has a coefficient of 1 on x^2.

vale moth
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yes

shut copper
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So we would want to factor out a like so; a(x^2 + (b/a)x + c/a)

vale moth
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yes okay

shut copper
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So now we have that 2a in our (x + a)^2 formula is equal to b/a in our quadratic.

vale moth
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okay

shut copper
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Um, let's instead say (x + y)^2 = x^2 + 2yx + y^2.

vale moth
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yes

shut copper
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So we don't get confused with different a's.

vale moth
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thank you

shut copper
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Then 2y = b/a, therefore y = b/2a, and y^2 = b^2/4a^2.

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So we want to add b^2/4a^2, but we can't, because that would change the value of the expression, so what do we have to do to fix that?

vale moth
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okay

shut copper
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...so what do we have to do to fix that?

vale moth
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um

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i have no idea

shut copper
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If I add an amount, what do I have to do to get back to the amount I had before?

vale moth
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minus it

shut copper
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Right.

vale moth
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OH

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i see what you mean now

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a little bit i think

shut copper
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So we have so far: ax^2 + bx + c = a(x^2 + (b/a)x + c/a) = a(x^2 + (b/a)x + b^2/4a^2 - b^2/4a^2 + c/a)

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We now have all the pieces we need to complete the square, giving us: ax^2 + bx + c = a(x^2 + (b/a)x + c/a) = a(x^2 + (b/a)x + b^2/4a^2 - b^2/4a^2 + c/a) = a((x + b/2a)^2 + c/a - b^2/4a^2)Then we simply factor a back in, giving: = a(x + b/2a)^2 + c - b^2/4a

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Now, here's a question; what's important about this form?

vale moth
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hm

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i have never seen something like this before

shut copper
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If we're interested in minima and maxima of the expression over varying values of x, what might be important here?

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Where is x in this expression?

vale moth
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inside the bracket

shut copper
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Right, and that's the only place it is. And what are we doing to the bracket?

vale moth
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multiplying it by a

shut copper
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...what else?

vale moth
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and its a square

shut copper
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Yes! And what does that mean?

vale moth
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or sorry

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that theres 2?

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wait

shut copper
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What happens to a number when we square it?

vale moth
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it squares

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wait

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it just

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it multiplys itself

shut copper
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What's a positive number squared?

vale moth
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a positive

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so is a negative

shut copper
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What's a negative number squared?

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And what's 0 squared?

vale moth
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posivtive

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o

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0

shut copper
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Thus, no matter what value of x we plug in, (x + b/2a)^2 is always nonnegative.

vale moth
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nonnegative meaning that it will never be a negative?

shut copper
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Yes.

vale moth
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yes that is true

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wait

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so the minimum is 0

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right?

shut copper
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No.

vale moth
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oh

shut copper
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Remember the entire expression is a(x + b/2a)^2 + c - b^2/4a.

vale moth
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okay

shut copper
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In fact, let's say f(x) = a(x + b/2a)^2 + c - b^2/4a, so we can refer to f(x) to save space.

vale moth
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okay

shut copper
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So when does f(x) have a minimum?

vale moth
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when?

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when it has a value

shut copper
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Yes.

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No.

vale moth
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oh

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hmm

shut copper
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When we talk about f(x) in the abstract, without referring to a particular value of x, we're talking about the general behavior of f over all possible values of x.

vale moth
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okay

shut copper
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So we know that (x + b/2a)^2 is nonnegative, right? So the smallest it can be is 0. What's the largest it can be?

vale moth
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infinite

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infinity

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any positive number right?

shut copper
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Right.

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So what effect does that have on f?

vale moth
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i apologise for my lack of maths terms

shut copper
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No, it's fine, what's important is that you're taking the time to think it through.

vale moth
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the effect is that f has to be a positive?

shut copper
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Not necessarily.

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Okay, think about it like this. We know that the smallest (x + b/2a)^2 can be is 0. What's the smallest a(x + b/2a)^2 can be?

vale moth
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any number?

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like including positives and negatives

shut copper
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Can it be any number ever?

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Is there anything that it depends on?

vale moth
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im not sure

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the difference between the 2 is a

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so one of them u multiply everything by a then

shut copper
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Is it possible for a(x + b/2a)^2 to be both positive and negative just by changing x?

vale moth
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no

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wait

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a multiply by everything?

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if a was a positive and the value in the brackets is a negative then it creates a negative

shut copper
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Are you trying to expand (x + b/2a)^2?

shut copper
vale moth
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well

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a positive by a negative is a negative right?

vale moth
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im trying to imagine it with positive and negative numbers

shut copper
vale moth
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its always a nonnegative

shut copper
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Why?

vale moth
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because the smallest min is 0

shut copper
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Why?

vale moth
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something to do with that

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wait

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i dont know.

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wait i genuinely dont think i know

shut copper
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Is x + b/2a always nonnegative?

vale moth
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OH

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BECAUSE ITS A SQUARE

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sorry

shut copper
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Yes.

vale moth
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sorry yes that was a silly mistake by me

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yes its nonnegative always because its a square

shut copper
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So if (x + b/2a)^2 is always nonnegative, what is the only thing that can affect the sign of a(x + b/2a)^2?

vale moth
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the a outside the bracket

shut copper
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And the value of a doesn't change, it's what we call a constant, which means a(x + b/2a)^2 is either always nonnegative or always nonpositive, depending on the sign of a.

vale moth
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oh

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sorry

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i didnt know that

shut copper
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You literally just knew that. You just figured that out.

vale moth
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wait what

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well yes now i know that if the value of a doesnt change from being a positive then its a nonnegative

shut copper
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No, no, no.

vale moth
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but if it doesnt change from being a negative its a nonpositive

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oh

shut copper
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Okay, wait, yes.

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When you said "if a doesn't change", I thought you meant that you thought a could change.

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No, only x changes.

vale moth
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i meant its a constant

shut copper
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Right.

vale moth
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so a is a constant but x can change

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depending on the sign of a, it can be a positive and a negative

shut copper
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x is the only thing that can change. If we change a, b, or c, we're talking about a completely different function.

vale moth
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okay

shut copper
vale moth
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a,b,c as in the coefficient of quadratic equations?

shut copper
vale moth
vale moth
vale moth
shut copper
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So now let's take it back to the original question, when does f(x) = a(x + b/2a)^2 + c - b^2/4a have a minimum?

vale moth
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i still dont really get what you mean by when unfortunately

shut copper
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For which values of a, b, c?

vale moth
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the whole idea of when

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like

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i think it has a minimum when its told in the question

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i dont see it at a deeper level than that

shut copper
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"When" means "for which combinations of values of the constants"?

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Or "for which combinations of values of the variables", depending.

vale moth
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for which combinations of values does f(x) = to the equation?

shut copper
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Which functions of the form f(x) = a(x + b/2a)^2 + c - b^2/4a have a minimum?

vale moth
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the a(x + b/2a)^2

shut copper
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No.

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Maybe it's break time.

vale moth
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your right

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its 11:50 for me

shut copper
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Go take a short break, like 15 minutes, to get a snack or a drink or something.

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Oh. Or, maybe go to bed, that sounds like a good idea too.

vale moth
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Your right

shut copper
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...about which idea?

vale moth
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going to bed i would say

shut copper
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Because if you go with the short break, I might be able to finish teaching you when you get back.

vale moth
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ill have a chat with my maths teacher

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i just wanna ask though

shut copper
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Honestly, your teacher kind of sounds like part of the problem if they didn't teach you the stuff I was trying to explain to you.

vale moth
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you said at the start u asked me for the simplified or the depper version

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has this whole time been the simplified version

shut copper
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Yeah.

vale moth
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something i should know that i don

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dont

shut copper
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...this whole conversation.

vale moth
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oof

shut copper
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That's what I was just saying.

vale moth
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but i just started max and min of quadratic graphs?

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im meant to know about it fully before doing it?

shut copper
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You're meant to know how to do your homework.

vale moth
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your right

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i got the smallest possible value 3

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i dont really even know how myself

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ill go ask my teacher though

shut copper
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I can explain using what we've learned so far.

vale moth
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explain how to get the smallest possible value?

shut copper
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Yes.

vale moth
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okay

shut copper
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So we're talking about f(x) = x^2 + 4x + 7, right?

vale moth
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yes

shut copper
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Let's convert it into the form we've been talking about. We have a = 1, b = 4, c = 7.

vale moth
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i completed the square and got (x+2)squared +3

vale moth
shut copper
vale moth
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okay

shut copper
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Now, we were just talking about how (x + 2)² is always nonnegative, right?

vale moth
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yes

shut copper
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Meaning, what's the least it can ever be?

vale moth
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0

shut copper
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And in this case, we find that a = 1 > 0.

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That's the key.

vale moth
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OH

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sorry

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yes i get you

shut copper
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When a > 0 it has a minimum, when a < 0 there's a maximum.

vale moth
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yes yes

shut copper
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So there's a minimum, and we get it when the squared term is the smallest it can ever be, which is 0.

vale moth
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yes

shut copper
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And what do we get when that bit is 0?

vale moth
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that bit as in when the smallest it can ever be is 0?

shut copper
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When the squared term is 0, what is the value of the whole function?

vale moth
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0

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0 squared is 0

shut copper
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So f(x) = (x + 2)² + 3, and when (x + 2)² = 0, you're saying f(x) = 0?

vale moth
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oh

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OHHH

shut copper
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So 0 + 3 = 0?

vale moth
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you not

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omg

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i see it

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its 3

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yes i apologise

shut copper
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Yes, because changing the value of x can only ever possibly increase the value of f.

vale moth
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its saying f(x) = 0 +3

shut copper
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From the minimum.

vale moth
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okay

shut copper
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So the minimum is the value you can't go down from.

vale moth
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yes that is correct

shut copper
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That's why we get the minimum when the squared term, which is always nonnegative, is 0.

vale moth
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i get it

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well

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yes

shut copper
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So now, when is (x + 2)² = 0?

vale moth
vale moth
shut copper
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And that's b.

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As for c, we're talking about a fraction, a division, right? What are two ways to make a/b bigger?

vale moth
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adding and multiplying?

shut copper
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No, when is a/b > c/b?

vale moth
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when a is bigger than c

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right?

shut copper
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Well.

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When a > c and b > 0, or when a < c and b < 0.

vale moth
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oh

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right

shut copper
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When is a/b > a/c?

vale moth
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when b is smaller

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like as in a half and a quarter

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if b was 2 and c was 3

shut copper
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We need to solve the inequality.

vale moth
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i dont really see how we would solve it

shut copper
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In the same way we solve equations, except being extra mindful of steps that might change the direction of inequality.

vale moth
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okay

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so

shut copper
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So what are the rules for solving equations?

vale moth
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hm

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i didnt know there were rules for solving equations

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i never thought of it that way

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i dont really have an answer

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solving means getting a value

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i would say

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when the questions says to solve i would think of it as its looking for a final value

shut copper
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a = b implies:
a + c = b + c
a * c = b * c
a - c = b - c
For all c
a / c = b / c
For all c =/= 0

vale moth
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yes

shut copper
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So a > b implies
a + c > b + c
a - c > b - c
For all c

a * c > b * c
For all c > 0
a * c < b * c
For all c < 0

a / c > b / c
For all c > 0
a / c < b / c
For all c < 0

vale moth
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?

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or no

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1/b > 1/c

shut copper
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If?

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How did you get there?

vale moth
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the a

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a is in both

shut copper
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So what did you do?

vale moth
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so i just removed it

shut copper
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How?

vale moth
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i just..

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cancelled

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i think

shut copper
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But how does that actually happen according to the rules?

vale moth
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because a is the same value in both?

shut copper
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No.

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If a/b = c, then 1/b equals what?

vale moth
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c

shut copper
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No.

vale moth
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it equals (-a)(c)

shut copper
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You're guessing.

vale moth
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i brang over the a

shut copper
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We don't guess in math. We know or we don't.

vale moth
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okay

shut copper
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So if you don't know, it's important to be able to admit that. And if you do know, it's important to be able to explain how.

vale moth
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but i thought you could bring over the a over the = sign and change the sign and division into multiplication

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how when u turn over a positive over the = sign it turns into a negative

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i tried to apply that

shut copper
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That's a misunderstanding of the rules I've told you.

vale moth
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oh

shut copper
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a/b = c implies (a/b)/a = c/a.

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(If a isn't 0)

vale moth
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you put them both over a

shut copper
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Yes.

vale moth
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arent both of those ways the same though?

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a/b = c the same as saying (a/b)/a = c/a

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is it not?

shut copper
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Yes, that's the point.

vale moth
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oh

shut copper
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An equation is a statement, an assertion of equality.

vale moth
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i thought we were looking for a final value

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i apologise

shut copper
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When we talk of "solving" an equation, we mean to find what values of the variables make the equation true.

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We do this by transforming the equation into new equations which are true if and only if the original equation is true.

vale moth
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okay

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i understand

shut copper
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I'm sorry, we've come a bit far afield.

vale moth
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no i needed this

shut copper
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We were trying to solve problem iii.

vale moth
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i never thought of it this way

vale moth
shut copper
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Few people do because it's not taught.

vale moth
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thank you

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at least now i can realise a little better what im actually looking at

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i think ill go to sleep now though because im genuinely half awake 😭

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ill most likely return tommorow if i dont get it at school tommorow

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thank you very much for having the patience to explain this to me

shut copper
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No problem. Thank you for listening, a surprising number of people don't.

And frankly I consider it a fundamental failure of the education system that you haven't been taught this. Algebra is fundamentally the mathematical study of equations, and it's a crime that students aren't even taught the fundamental fact that an equation is a statement. That fact in itself informs basically everything about algebra! When you talk about graphing a function as y = f(x), what you're actually doing is marking all the points (x, y) in the plane where the equation y = f(x) is true.

gilded sageBOT
#

@vale moth

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vale moth
#

+close

gilded sageBOT
# vale moth +close
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# gilded sage

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