8th grade math, algebraic fractions and formulae | Mathematics | Page 1

uneven lodge Jul 7, 2025, 3:28 AM

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nimble parcelBOT Jul 7, 2025, 3:28 AM

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uneven lodge

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uneven lodge Jul 7, 2025, 3:29 AM

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need help with a) and b), and maybe some help understanding this topic overall. thanks in advance

hidden holly Jul 7, 2025, 3:29 AM

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Okay, so.

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When we talk about doing algebra.

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What we're really doing is transforming equations into other equations.

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And we're doing this in specific ways.

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Specifically, an equation is a statement.

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Therefore it has a truth value.

uneven lodge Jul 7, 2025, 3:30 AM

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whats a truth value

hidden holly Jul 7, 2025, 3:30 AM

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True or false.

uneven lodge Jul 7, 2025, 3:31 AM

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i see

hidden holly Jul 7, 2025, 3:31 AM

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And we want the new equation to have the same truth value as the old one.

uneven lodge Jul 7, 2025, 3:31 AM

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ah i wasnt really taught this way

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in school and all

hidden holly Jul 7, 2025, 3:32 AM

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Right, so.

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The rules are, if a = b, then: a + c = b + c a - c = b - c ac = bc a/c = b/c (if c =/= 0)

uneven lodge Jul 7, 2025, 3:33 AM

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ohh i see

hidden holly Jul 7, 2025, 3:34 AM

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uneven lodge

So we begin with the equation (3a + 2)/20 = 5/(3a + 2)

uneven lodge Jul 7, 2025, 3:34 AM

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mhm

hidden holly Jul 7, 2025, 3:35 AM

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What we want to do is find the value of a that makes the equation true.

uneven lodge Jul 7, 2025, 3:35 AM

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alright

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was cross multiplying the first step correct?

hidden holly Jul 7, 2025, 3:36 AM

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Like I said, we can add, subtract, multiply, or divide the same thing on both sides.

uneven lodge Jul 7, 2025, 3:36 AM

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oh yeahh

hidden holly Jul 7, 2025, 3:37 AM

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So what should we do on both sides?

uneven lodge Jul 7, 2025, 3:37 AM

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in my opinion maybe make them like not be a fraction?

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dont really know how to phrase it

hidden holly Jul 7, 2025, 3:38 AM

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How do we do that?

uneven lodge Jul 7, 2025, 3:38 AM

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cross multiplying it?

hidden holly Jul 7, 2025, 3:38 AM

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Again, we have to multiply the same thing on both sides.

uneven lodge Jul 7, 2025, 3:39 AM

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3a+2/20 times 20 = 5/3a+2 times 20?

hidden holly Jul 7, 2025, 3:40 AM

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Right.

uneven lodge Jul 7, 2025, 3:40 AM

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it'll remove the denominator from 3a+2/20

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then uhh 5/3a+ 2 times 20 times 3a+2

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(3a+2)^2

hidden holly Jul 7, 2025, 3:40 AM

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(3a + 2)/20 = 5/(3a + 2) if and only if (3a + 2)/20 * 20 = 5/(3a + 2) * 20.

uneven lodge Jul 7, 2025, 3:41 AM

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gimmie a moment to absorb what u said

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oh alright

hidden holly Jul 7, 2025, 3:42 AM

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Remember, x = y if and only if zx = zy. Here, x = (3a + 2)/20, y = 5/(3a + 2), z = 20.

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Now let's simplify this equation before moving on.

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We get 3a + 2 = 100/(3a + 2), right?

uneven lodge Jul 7, 2025, 3:44 AM

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yep

hidden holly Jul 7, 2025, 3:45 AM

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So what should we do next?

uneven lodge Jul 7, 2025, 3:46 AM

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100/(3a+2) times (3a+2)

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(3a+2)^2 = 100

hidden holly Jul 7, 2025, 3:47 AM

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Right. 3a + 2 = 100/(3a + 2) if and only if (3a + 2)^2 = 100.

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Now, another rule is that a^2 = b^2 if and only if a = b or a = -b.

uneven lodge Jul 7, 2025, 3:49 AM

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the last part

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a= -b

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i dont quite understand it

hidden holly Jul 7, 2025, 3:49 AM

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Because what's (-b)^2?

uneven lodge Jul 7, 2025, 3:49 AM

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ohh

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positive b suqare

hidden holly Jul 7, 2025, 3:49 AM

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Right.

uneven lodge Jul 7, 2025, 3:49 AM

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-b x -b

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alright thanks i understand that part now

hidden holly Jul 7, 2025, 3:50 AM

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So we have 3a + 2 = 10 or 3a + 2 = -10.

uneven lodge Jul 7, 2025, 3:50 AM

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wait what

hidden holly Jul 7, 2025, 3:51 AM

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It could be either.

uneven lodge Jul 7, 2025, 3:51 AM

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hidden holly So we have `3a + 2 = 10` *or* `3a + 2 = -10`.

no i mean like how did u get this

hidden holly Jul 7, 2025, 3:51 AM

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Because (3a + 2)^2 = 100.

uneven lodge Jul 7, 2025, 3:51 AM

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oh u sq rooted it?

hidden holly Jul 7, 2025, 3:52 AM

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In a way.

uneven lodge Jul 7, 2025, 3:52 AM

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i see

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wait why can it be -10 though

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or just negative in general

hidden holly Jul 7, 2025, 3:53 AM

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Because if 3a + 2 = -10, then (3a + 2)^2 = 100.

uneven lodge Jul 7, 2025, 3:53 AM

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oh wait yeah i keep forgetting negative multiplied by negative is positive

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alright thanks

hidden holly Jul 7, 2025, 3:54 AM

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So now we have two equations, and we have to solve both, and then check the solutions.

uneven lodge Jul 7, 2025, 3:54 AM

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check the solutions?

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sorry for not understanding certain terms, the education system here is quite different in a sense

hidden holly Jul 7, 2025, 3:55 AM

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Yes. It's possible we get what are called "extraneous" solutions.

uneven lodge Jul 7, 2025, 3:55 AM

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hidden holly Yes. It's possible we get what are called "extraneous" solutions.

i've never heard of that term before

hidden holly Jul 7, 2025, 3:55 AM

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Remember what I said about transforming equations into different equations with the same truth value?

uneven lodge Jul 7, 2025, 3:55 AM

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yeah

hidden holly Jul 7, 2025, 3:56 AM

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That doesn't always exactly work.

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Sometimes we transform an equation into an equation with more solutions than the original had.

uneven lodge Jul 7, 2025, 3:57 AM

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i see

hidden holly Jul 7, 2025, 3:57 AM

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So we have to take the solutions we get and plug them back into the original equation to make sure they really are solutions to that equation.

uneven lodge Jul 7, 2025, 3:58 AM

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ohh its like checking if the answer is correct?

hidden holly Jul 7, 2025, 3:58 AM

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Yes.

uneven lodge Jul 7, 2025, 3:58 AM

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oh i see i think i did something similar to that when i was learning simultaneous equations

hidden holly Jul 7, 2025, 3:59 AM

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hidden holly So we have `3a + 2 = 10` *or* `3a + 2 = -10`.

So, returning to here, which equation would you like to solve first?

uneven lodge Jul 7, 2025, 3:59 AM

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probably the positive one

hidden holly Jul 7, 2025, 3:59 AM

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So we have 3a + 2 = 10.

uneven lodge Jul 7, 2025, 4:00 AM

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hidden holly So we have `3a + 2 = 10` *or* `3a + 2 = -10`.

3a +2 -2= 10-2
3a= 8
a= 8/3

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is that right?

hidden holly Jul 7, 2025, 4:00 AM

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And now we check, is it true that, if a = 8/3, then (3a + 2)/20 = 5/(3a + 2)?

uneven lodge Jul 7, 2025, 4:01 AM

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alright i'll grab my calc

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(3x8/3+2) divided by 20= 0.5
5/(3x8/3+2)= 50

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its false i think

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or i made some mistakes while inputting the numbers in my calc

hidden holly Jul 7, 2025, 4:06 AM

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I think you made a mistake.

uneven lodge Jul 7, 2025, 4:06 AM

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i see i'll try calculating it again

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mistake in "a=8/3" or calculation mistake?

hidden holly Jul 7, 2025, 4:09 AM

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Calculator mistake.

uneven lodge Jul 7, 2025, 4:09 AM

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hidden holly And now we check, is it *true* that, *if* `a = 8/3`, *then* `(3a + 2)/20 = 5/(3a...

(3a+2)/20 is
3a= 3 times 8/3
3a = 8
8+2=10
10/20= 0.5

5/(3a + 2) is
3a= 3 times 8/3
3a= 8
8+2= 10
5 divided by 10= 0.5

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wait i gotta edit it

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oh its divide mb wait

uneven lodge Jul 7, 2025, 4:11 AM

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uneven lodge (3a+2)/20 is 3a= 3 times 8/3 3a = 8 8+2=10 10/20= 0.5 5/(3a + 2) is 3a= 3 time...

alright yeah its true

hidden holly Jul 7, 2025, 4:13 AM

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Great. Now we solve the other equation.

uneven lodge Jul 7, 2025, 4:13 AM

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back

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3a + 2-2 = -10 -2
3a= -12
a= -4

hidden holly Jul 7, 2025, 4:15 AM

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And now we check that solution.

uneven lodge Jul 7, 2025, 4:16 AM

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alright

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-0.5 = -0.5

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its true

hidden holly Jul 7, 2025, 4:17 AM

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So both of our solutions are solutions.

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Do you think you can solve b on your own?

uneven lodge Jul 7, 2025, 4:18 AM

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mm

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are they both similar?

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if yes then i'll give it a shot

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thanks for the in-depth explanations

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have a great day

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_goobertea

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i'll leave this open for now in case i cant solve B

hidden holly Jul 7, 2025, 4:20 AM

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uneven lodge i'll leave this open for now in case i cant solve B

You could show your work in the thread if you like.

uneven lodge Jul 7, 2025, 4:22 AM

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alright I'll show it when Ive finished it and y'all can "mark" it since my teacher is on holiday

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like verify is my answers are right or wrong

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or i could just check the solution

uneven lodge Jul 7, 2025, 5:06 AM

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im unsure on how to do b)

uneven lodge Jul 7, 2025, 5:06 AM

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uneven lodge

working done is still the same

hidden holly Jul 7, 2025, 5:07 AM

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Okay, let's think about it. We begin with the equation 36/x - x = 5, correct?

uneven lodge Jul 7, 2025, 5:07 AM

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yep

hidden holly Jul 7, 2025, 5:12 AM

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Now we can do the same thing to both sides of the equation. What should we do?

uneven lodge Jul 7, 2025, 5:16 AM

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back

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uhh

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is it 36/x times x - x times x = 5(x)?

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36-x^2= 5x

hidden holly Jul 7, 2025, 5:18 AM

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That is true if the original equation is true.

uneven lodge Jul 7, 2025, 5:20 AM

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wait

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my friend did say i forget to do something

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other qn u forgot times an x on the rhs
jaydernn
[ROLI]
— Yesterday at 12:40 PM
what
Greg 👍
[ASTD]
— Yesterday at 12:40 PM
becuz urfirstequation times an x first
but u nvr times the x to the rhs

hidden holly Jul 7, 2025, 5:21 AM

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I mean.

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That's from yesterday.

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Let's talk about the work you're doing now.

uneven lodge Jul 7, 2025, 5:22 AM

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alright

uneven lodge Jul 7, 2025, 5:22 AM

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uneven lodge 36-x^2= 5x

is this correct?

hidden holly Jul 7, 2025, 5:24 AM

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uneven lodge is this correct?

Yes. You multiplied the same thing on both sides.

uneven lodge Jul 7, 2025, 5:24 AM

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alright'

uneven lodge Jul 7, 2025, 5:24 AM

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uneven lodge 36-x^2= 5x

so i need to solve for x

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x= 36-x^2/5?

hidden holly Jul 7, 2025, 5:25 AM

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You can't have x on both sides.

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Also you mean (36 - x^2)/5.

uneven lodge Jul 7, 2025, 5:26 AM

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hidden holly Also you mean `(36 - x^2)/5`.

oh yeahh sorry for the mistake

uneven lodge Jul 7, 2025, 5:26 AM

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hidden holly You can't have x on both sides.

what does this mean?

hidden holly Jul 7, 2025, 5:26 AM

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uneven lodge what does this mean?

It means we want to know what number x is equal to.

uneven lodge Jul 7, 2025, 5:26 AM

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ohh

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oh i see so x cant be repeated on the other side

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so 36-x^2=5x

hidden holly Jul 7, 2025, 5:27 AM

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Right, because then we can only know what number x is if... we know what number x is.

hidden holly Jul 7, 2025, 5:28 AM

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uneven lodge so 36-x^2=5x

Right.

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Now, this is what we call a quadratic equation, and there are a few different ways to solve it (that are really the same way).

uneven lodge Jul 7, 2025, 5:28 AM

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hidden holly Now, this is what we call a quadratic equation, and there are a few different wa...

i see

hidden holly Jul 7, 2025, 5:29 AM

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Here's one way we call "completing the square".

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First, we need to get all our x terms on one side.

uneven lodge Jul 7, 2025, 5:29 AM

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36-x^2-5x=5x-5x

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36-x^2-5x=0?

hidden holly Jul 7, 2025, 5:30 AM

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We need x terms on one side, numbers on the other.

uneven lodge Jul 7, 2025, 5:30 AM

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ohh

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(36-x^2)x=5?

hidden holly Jul 7, 2025, 5:32 AM

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...no.

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Look.

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36 - x^2 = 5x
36 - x^2 + x^2 = 5x + x^2
x^2 + 5x = 36```

uneven lodge Jul 7, 2025, 5:33 AM

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ohh i see so now all the x terms are on one side

hidden holly Jul 7, 2025, 5:34 AM

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Yes.

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Now, here's a question.

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What's (a + b)^2?

uneven lodge Jul 7, 2025, 5:34 AM

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a+ 2ab + b

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i mean

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a sq + 2ab + b sq

hidden holly Jul 7, 2025, 5:36 AM

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Right.

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So (x + y)^2 = x^2 + 2yx + y^2, right?

uneven lodge Jul 7, 2025, 5:36 AM

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yep

hidden holly Jul 7, 2025, 5:36 AM

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So if 2yx = 5x, what happens when we solve for y?

uneven lodge Jul 7, 2025, 5:37 AM

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divide 2yx by 2y

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5x/2y?

hidden holly Jul 7, 2025, 5:37 AM

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...no.

uneven lodge Jul 7, 2025, 5:37 AM

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i mean 2x

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divide 2yx by 2x

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5x/2x?

hidden holly Jul 7, 2025, 5:38 AM

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Therefore?

uneven lodge Jul 7, 2025, 5:38 AM

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2.5x?

hidden holly Jul 7, 2025, 5:39 AM

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...no.

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What's x/x?

uneven lodge Jul 7, 2025, 5:40 AM

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0

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2.5

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so its 2.5

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?

hidden holly Jul 7, 2025, 5:40 AM

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Right, y = 2.5.

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So now, given x^2 + 5x = 36, we add y^2 on both sides.

uneven lodge Jul 7, 2025, 5:41 AM

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wait

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where did the y come from? or was it just an example

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sorry im really confused right now

hidden holly Jul 7, 2025, 5:43 AM

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(x + y)^2 = x^2 + 2yx + y^2.

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We have x^2 + 5x = 36.

uneven lodge Jul 7, 2025, 5:43 AM

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yeah the algebraic identity

hidden holly Jul 7, 2025, 5:43 AM

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It's called "completing the square" because we add the thing to both sides that we need to get a perfect square on the left side.

uneven lodge Jul 7, 2025, 5:44 AM

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oh

hidden holly Jul 7, 2025, 5:44 AM

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If y = 2.5, then we have x^2 + 2 * 2.5 * x + 2.5^2 = (x + 2.5)^2.

uneven lodge Jul 7, 2025, 5:45 AM

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wait sorry i have to eat lunch now

hidden holly Jul 7, 2025, 5:45 AM

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I might not be here when you get back, it's late for me.

uneven lodge Jul 7, 2025, 5:45 AM

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and i'll be away so yeah, thanks for the help for now

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its okay, get some rest. maybe my teacher can help me too

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thanks for the help

warm mantleBOT Jul 8, 2025, 6:00 AM

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@uneven lodge

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@hidden holly The user still needs help with this help request.

uneven lodge Jul 9, 2025, 12:02 PM

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+close

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warm mantleBOT Jul 9, 2025, 12:02 PM

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warm mantle

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uneven lodge Jul 9, 2025, 12:02 PM

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thanks'!

uneven lodge Jul 9, 2025, 12:28 PM

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is y=2 in this case?

uneven lodge Jul 9, 2025, 12:51 PM

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hidden holly So now we have *two* equations, and we have to solve both, and then check the so...

i've checked my solutions and it seems to be wrong. y=2 is false

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since if y=2, the -1/y-2 part would essentially be -1/0

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math error

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so yeah

hidden holly Jul 9, 2025, 8:03 PM

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uneven lodge i've checked my solutions and it seems to be wrong. y=2 is false

You made a mistake.

uneven lodge Jul 10, 2025, 5:30 AM

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where was the mistake

hidden holly Jul 10, 2025, 6:36 AM

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uneven lodge where was the mistake

First line to second line.

coral torrent Jul 10, 2025, 6:36 AM

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Very smart

hidden holly Jul 10, 2025, 6:36 AM

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Also, more generally, I think until you have a more solid intuition that you should do it explicitly and exactly the way I taught you; take the equation and do the same thing on both sides.

uneven lodge Jul 10, 2025, 1:25 PM

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I see alright thanks

#8th grade math, algebraic fractions and formulae