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March 28, 2024, 11:58:01 pm

Author Topic: how do I factor a quadratic equation?  (Read 1951 times)  Share 

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annaoh_2003

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how do I factor a quadratic equation?
« on: January 23, 2022, 05:03:26 pm »
+1
Hi everyone,

Im wanting to find x intercepts and wondering how to factor this equation. I used an algebra calculator but it didn't tell me how to factor the equation.

Thanks :)

Tapioca

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Re: how do I factor a quadratic equation?
« Reply #1 on: January 23, 2022, 05:10:17 pm »
+1
.
« Last Edit: March 11, 2022, 12:28:02 am by Tapioca »

annaoh_2003

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Re: how do I factor a quadratic equation?
« Reply #2 on: January 23, 2022, 05:18:43 pm »
+1
Let f(x) = 2x^3 - 3x^2 -3x + 2
Because f(-1)= 0, then x+1 is a factor:

Factorise
= 2x^2(x+1) - 5x(x+1) +2(x+1)

Simplify
= (x+1)(2x^2 -5x +2)
= (x+1) (2x-1) (x-2)

Done  :)


little bit confused- how come you used -1 in f(x)? im struggling to see how you factorised and simplified it. sorry im a bit slow with maths  ;D could you explain it a bit more?

Tapioca

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Re: how do I factor a quadratic equation?
« Reply #3 on: January 23, 2022, 05:30:36 pm »
+1
.
« Last Edit: March 11, 2022, 12:28:09 am by Tapioca »

Bri MT

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Re: how do I factor a quadratic equation?
« Reply #4 on: January 23, 2022, 06:17:30 pm »
0

little bit confused- how come you used -1 in f(x)? im struggling to see how you factorised and simplified it. sorry im a bit slow with maths  ;D could you explain it a bit more?

I'm adding to Tapioca's explanation since it seems like you may not have covered the polynomial theorem in school yet which teaches that if f(a) = 0 then (x -a) is a factor of f(x). It's common to test easy numbers first (like 1, -1, 2, -2) when trying to find the first factor.

It might be easiest to first see this looking from the other perspective.
if you have (x-a)(x-c)(x-d)  then we know we can make the whole thing = 0 by making one of the brackets equal zero. (null factor law).
I.e. if (x-a) is a factor, then making x = a means that the equation equals zero.

So to restate, Tapioca was using the similar idea that if have x = a makes the whole thing equal zero, then that means (x-a) is a factor.

That's the first step.  The 2nd step may also be a bit tricky but before we move on I think it's important to check that you get the first one :)


Kudos to you for reaching out for help and to Tapioca for the explanations :)

annaoh_2003

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Re: how do I factor a quadratic equation?
« Reply #5 on: January 23, 2022, 06:55:12 pm »
0
I'm adding to Tapioca's explanation since it seems like you may not have covered the polynomial theorem in school yet which teaches that if f(a) = 0 then (x -a) is a factor of f(x). It's common to test easy numbers first (like 1, -1, 2, -2) when trying to find the first factor.

It might be easiest to first see this looking from the other perspective.
if you have (x-a)(x-c)(x-d)  then we know we can make the whole thing = 0 by making one of the brackets equal zero. (null factor law).
I.e. if (x-a) is a factor, then making x = a means that the equation equals zero.


Let f(x) = 2x^3 - 3x^2 -3x + 2
Because f(-1)= 0, then x+1 is a factor:


Since you wanted to find the X-intercepts, we know that y=0:

0= (x-1) (2x-1) (x-2)

So,
(x-1)=0
(2x-1)=0
(x-2)=0

From algebraically solving, the X-intercepts: x=1 and x=1/2 and x=2

ohhh wait so my brain just clicked- you're actually using the -1 in place of the x - okay got it now  :)

so when you factorise how did you find = 2x^2(x+1) - 5x(x+1) +2(x+1) ? I can see from working it out we get our original equation - but how did you work backwards to find those values in red?

Tapioca

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Re: how do I factor a quadratic equation?
« Reply #6 on: January 23, 2022, 07:27:11 pm »
+1
.
« Last Edit: March 11, 2022, 12:28:19 am by Tapioca »

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Re: how do I factor a quadratic equation?
« Reply #7 on: January 23, 2022, 07:43:01 pm »
+1
ohhh wait so my brain just clicked- you're actually using the -1 in place of the x - okay got it now  :)

so when you factorise how did you find = 2x^2(x+1) - 5x(x+1) +2(x+1) ? I can see from working it out we get our original equation - but how did you work backwards to find those values in red?

I failed methods myself but this is my way of going about the factorisation to get the quadratic component: https://drive.google.com/file/d/106qGlePQrAL0OAmtfH_RgROkOKwYN45q/view?usp=drivesdk
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annaoh_2003

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Re: how do I factor a quadratic equation?
« Reply #8 on: January 23, 2022, 08:38:26 pm »
+1

About the factorising step- you are basically taking (x+1) out of the equation. Take this out of each term, then think about how to get to the original equation each time you factorise a term, then work your way to the end. After this collect the terms outside brackets with positives and negatives, then factorise to simplest form.

I'm adding to Tapioca's explanation since it seems like you may not have covered the polynomial theorem in school yet which teaches that if f(a) = 0 then (x -a) is a factor of f(x). It's common to test easy numbers first (like 1, -1, 2, -2) when trying to find the first factor.

It might be easiest to first see this looking from the other perspective.
if you have (x-a)(x-c)(x-d)  then we know we can make the whole thing = 0 by making one of the brackets equal zero. (null factor law).
I.e. if (x-a) is a factor, then making x = a means that the equation equals zero.

So to restate, Tapioca was using the similar idea that if have x = a makes the whole thing equal zero, then that means (x-a) is a factor.

That's the first step.  The 2nd step may also be a bit tricky but before we move on I think it's important to check that you get the first one :)


Kudos to you for reaching out for help and to Tapioca for the explanations :)

thank you guys ! im doing an intensive six week method course at unimelb and its extremely difficult + I didn't have any methods background knowledge. im just relying on the atarnotes community at the moment (you can see ive been asking quite a few questions on the methods thread) and im extremely grateful for all your help :)