Completing the square practice review | Mathematics | Page 1

stable plover May 1, 2024, 4:15 PM

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I am practicing completing the square, in this example I needed to find the roots of the equation y=x^2+6x+9 and I used two different methods as part of my practice. I verified the equation in an online graphing calculator (attached as image). I spelled out each step a lot as I want to practice my understanding thoroughly.

Im asking for a review in my approach, can you verify I did everything correctly? Do you have any suggestions or critique? Thanks for the help!

Solve for x to find the roots of parabola y=x^2+6x+9.

METHOD 1:
y=x^2+6x+9
x^2+6x+9=0
x^2+6x=-9

PLUS:
x^2+3^2+6x=-9+(-3)^2
(a+b)^2=a^2+b^2+2ab
x^2+3^2+2*2*3x=0
(x+3)^2=0
sqrt((x+3)^2)=sqrt(0)
x+3=0
x=-3

MINUS:
x^2-3^2+6x=-9-(-3)^2
(a-b)^2=a^2+b^2-2ab
x^2+3^2-2*2*3x=-18
(x-3)^2=-18
NO SOLUTION FOR MINUS, SINCE SQUARE ROOT OF -18 IS NOT POSSIBLE
NOT POSSIBLE: sqrt((x-3)^2)=sqrt(-18)

Therefore x=-3 is the only solution to solve for x in y=x^2+6x+9, and therefore y=x^2+6x+9 has only one root (-3,0) which is also the coordinate of the vertex of the parabola y=x^2+6x+9



METHOD 2:
$ax^2+bx+c=0$
$x2+(b/a)x=-c/a$
$(x+b/(2a))^2=-c/a+b^2/(4a^2)$
-c/a+b^2/(4a^2)=(b^2-4ac)/(4a^2)
x+b/(2a)=+squareroot(b^2-4ac)/(2a)
PLUS: x=(-6+squareroot(b^2-4ac))/(2a)
MINUS: x=(-6-squareroot(b^2-4ac))/(2a)

PLUS:
x=(-6+squareroot(b^2-4ac))/(2a)
x^2+6x+9=0
x=(-6+squareroot(6^2-4*9))/2
x=(-6+squareroot(36-36))/2
x=-6/2
x=-3

MINUS:
x=(-6-squareroot(b^2-4ac))/(2a)
x^2+6x+9=0
x=(-6-squareroot(6^2-4*9))/2
x=(-6-squareroot(36-36))/2
x=-6/2
x=-3

Therefore x=-3 is the only solution to solve for x in y=x^2+6x+9, and therefore y=x^2+6x+9 has only one root (-3,0) which is also the coordinate of the vertex of the parabola y=x^2+6x+9

proven valveBOT May 1, 2024, 4:15 PM

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stable plover May 1, 2024, 4:21 PM

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My apology, I forgot to format my equations for Discord, that may have messed up some symbols and made it seem wrong. I put in in code format now, so it should be correct now. Please discard the initial copy you did

hearty kayak May 1, 2024, 4:22 PM

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Your first two methods are right
But it's really difficult to read and understand. Better to quote in latex.

stable plover May 1, 2024, 4:23 PM

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Yes it is difficult, my text editor was limited.

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Is latex a program?

hearty kayak May 1, 2024, 4:23 PM

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No just put $ sign on both sides of an expression

stable plover May 1, 2024, 4:24 PM

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Doesn't seem to work

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I mean it still ends up the same

hearty kayak May 1, 2024, 4:24 PM

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That's what I know
It just formats it like
$x^2 + y^2 = 2^{a+b}$

neon vineBOT May 1, 2024, 4:25 PM

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autocrazizt

stable plover May 1, 2024, 4:25 PM

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hearty kayak May 1, 2024, 4:26 PM

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copy paste as text.
It's a the bot that transliterates for us

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I too learned recently

stable plover May 1, 2024, 4:26 PM

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$y=x^2+6x+9$
$x^2+6x+9=0$
$x^2+6x=-9$

neon vineBOT May 1, 2024, 4:26 PM

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DisplayName

stable plover May 1, 2024, 4:26 PM

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oh

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like that,

hearty kayak May 1, 2024, 4:26 PM

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Seems great, right?

stable plover May 1, 2024, 4:28 PM

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METHOD 1:
$y=x^2+6x+9$
$x^2+6x+9=0$
$x^2+6x=-9$

PLUS:
$x^2+3^2+6x=-9+(-3)^2$
$(a+b)^2=a^2+b^2+2ab$
$x^2+3^2+223x=0$
$(x+3)^2=0$
$sqrt((x+3)^2)=sqrt(0)$
$x+3=0$
$x=-3$

MINUS:
$x^2-3^2+6x=-9-(-3)^2$
$(a-b)^2=a^2+b^2-2ab$
$x^2+3^2-223x=-18$
$(x-3)^2=-18$
NO SOLUTION FOR MINUS, SINCE SQUARE ROOT OF $-18$ IS NOT POSSIBLE
NOT POSSIBLE: $sqrt((x-3)^2)=sqrt(-18)$

Therefore x=-3 is the only solution to solve for $x$ in $y=x^2+6x+9$, and therefore $y=x^2+6x+9$ has only one root $(-3,0)$ which is also the coordinate of the vertex of the parabola $y=x^2+6x+9$

METHOD 2:
$ax^2+bx+c=0$
$x2+(b/a)x=-c/a$
$(x+b/(2a))^2=-c/a+b^2/(4a^2)$
$-c/a+b^2/(4a^2)=(b^2-4ac)/(4a^2)$
$x+b/(2a)=+squareroot(b^2-4ac)/(2a)$
$PLUS: x=(-6+squareroot(b^2-4ac))/(2a)$
MINUS: $x=(-6-squareroot(b^2-4ac))/(2a)$

PLUS:
$x=(-6+squareroot(b^2-4ac))/(2a)$
$x^2+6x+9=0$
$x=(-6+squareroot(6^2-4*9))/2$
$x=(-6+squareroot(36-36))/2$
$x=-6/2$
$x=-3$

MINUS:
$x=(-6-squareroot(b^2-4ac))/(2a)$
$x^2+6x+9=0$
$x=(-6-squareroot(6^2-4*9))/2$
$x=(-6-squareroot(36-36))/2$
$x=-6/2$
$x=-3$

Therefore $x=-3$ is the only solution to solve for $x$ in $y=x^2+6x+9$, and therefore $y=x^2+6x+9$ has only one root $(-3,0)$ which is also the coordinate of the vertex of the parabola $y=x^2+6x+9$

neon vineBOT May 1, 2024, 4:29 PM

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DisplayName

stable plover May 1, 2024, 4:29 PM

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yikes

hearty kayak May 1, 2024, 4:29 PM

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Looks like it was better to snd it in bits

stable plover May 1, 2024, 4:29 PM

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It seems to discard some spacing and paragraphing formats that would have helped to keep

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METHOD 1:
$y=x^2+6x+9$
$x^2+6x+9=0$
$x^2+6x=-9$

neon vineBOT May 1, 2024, 4:29 PM

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DisplayName

stable plover May 1, 2024, 4:29 PM

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METHOD 1:
$y=x^2+6x+9$
$x^2+6x+9=0$
$x^2+6x=-9$

PLUS:
$x^2+3^2+6x=-9+(-3)^2$
$(a+b)^2=a^2+b^2+2ab$
$x^2+3^2+223x=0$
$(x+3)^2=0$
$sqrt((x+3)^2)=sqrt(0)$
$x+3=0$
$x=-3$

MINUS:
$x^2-3^2+6x=-9-(-3)^2$
$(a-b)^2=a^2+b^2-2ab$
$x^2+3^2-223x=-18$
$(x-3)^2=-18$
NO SOLUTION FOR MINUS, SINCE SQUARE ROOT OF $-18$ IS NOT POSSIBLE
NOT POSSIBLE: $sqrt((x-3)^2)=sqrt(-18)$

neon vineBOT May 1, 2024, 4:30 PM

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DisplayName

fading juniper May 1, 2024, 4:30 PM

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Bruh all of this to find roots of 2nd degree equation

stable plover May 1, 2024, 4:30 PM

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Maybe there is a way to format spacing into the latex?

hearty kayak May 1, 2024, 4:30 PM

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yeah there is,
I have asked in the help channel

stable plover May 1, 2024, 4:30 PM

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fading juniper Bruh all of this to find roots of 2nd degree equation

If you are here to help I would be happy if you could provide some help 🙂 as I mentioned in the OP Im spelling it out because I want to practice a thorough understanding of completing the square

fading juniper May 1, 2024, 4:31 PM

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not sure if I can read it ALL

hearty kayak May 1, 2024, 4:32 PM

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@fading juniper u know how to go to next line and add spacing in latex?

stable plover May 1, 2024, 4:32 PM

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fading juniper not sure if I can read it ALL

There's a format issue that Im working on resolving. The Latex seems to work great, though it needs some spacing and paragraphs to have better overview.

fading juniper May 1, 2024, 4:33 PM

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not realy I don't use LATEX at all

hearty kayak May 1, 2024, 4:33 PM

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I too prefer handwritten.

stable plover May 1, 2024, 4:34 PM

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I guess I could do that too and take an image with my phone and upload it

fading juniper May 1, 2024, 4:34 PM

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Latex is the best option if you are writing documents

stable plover May 1, 2024, 4:35 PM

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Thanks for the help guys 🙂

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im closing topic

hearty kayak May 1, 2024, 4:36 PM

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If you really want to practice try different questions rather than one. @stable plover

stable plover May 1, 2024, 4:36 PM

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hearty kayak If you really want to practice try different questions rather than one. <@339802...

How do you mean different questions?

hearty kayak May 1, 2024, 4:36 PM

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BTB Here's some documentation on latex

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https://www.overleaf.com/learn/latex/Mathematical_expressions

Mathematical expressions

An online LaTeX editor that’s easy to use. No installation, real-time collaboration, version control, hundreds of LaTeX templates, and more.

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Your methods are fine, Try different quadratic equations.

stable plover May 1, 2024, 4:38 PM

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hearty kayak Your methods are fine, Try different quadratic equations.

oh of course yes 🙂 Good idea, I am doign that too, this was just an example one where the equation was simple which allowed me to go into a lot of detail with it to understand it better.

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'

hearty kayak May 1, 2024, 4:39 PM

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You will strengthen your concept only through practice
I too used to try different methods to see if the answer comes the same, out of curiosity

#Completing the square practice review