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Not Dav

Member

From: Animal kingdom

Registered: 2021-04-22

Posts: 55

How to complete the square with x² having negative coefficient

I struggle completing the square without using the quadratic equation. Could some show me how does factoring out the negative works and how to find the vertex. A example problem is y= -x² +4x -3

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Mathegocart

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Registered: 2012-04-29

Posts: 2,226

Re: How to complete the square with x² having negative coefficient

This is relatively easy, all things considered. Divide the whole equation by -1 so the x^2 coefficient is positive 1, then add 3 to both sides, and boom, just plug and chug.

Last edited by Mathegocart (2021-04-23 07:11:38)

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Not Dav

Member

From: Animal kingdom

Registered: 2021-04-22

Posts: 55

Re: How to complete the square with x² having negative coefficient

I'll try that thank you!

Last edited by Not Dav (2021-04-23 07:10:11)

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Not Dav

Member

From: Animal kingdom

Registered: 2021-04-22

Posts: 55

Re: How to complete the square with x² having negative coefficient

I was factoring the -1 out which made the problem very difficult instead of dividing on both sides. Lol

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mathland

Member

Registered: 2021-03-25

Posts: 444

Re: How to complete the square with x² having negative coefficient

I struggle completing the square without using the quadratic equation. Could some show me how does factoring out the negative works and how to find the vertex. A example problem is y= -x² +4x -3

y = -x² +4x -3

y= (-x² +4x -3)/(-1)

y = x²  - 4x + 3

We can now complete the square.

Set equation to 0.

x²  - 4x + 3 = 0

x²  - 4x = -3

Divide the middle term by 2 and raise to the second power.

[(-4/2)]^2 = (-2)^2 = 4.

Add 4 to both sides.

x²  - 4x + 4 = -3 + 4

x²  - 4x + 4 = 1

The left side becomes (x - 2)(x - 2) or simply (x - 2)^2.

We now have this:

(x - 2)^2 = 1

Take the square root on both sides.

sqrt{(x - 2)^2} = sqrt{1}

x - 2 = 1

x = 1 + 2

x = 3

And

x - 2 = -1

x = -1 + 2

x = 1

Done.

Last edited by mathland (2021-04-24 07:47:10)

Not Dav

Member

From: Animal kingdom

Registered: 2021-04-22

Posts: 55

Re: How to complete the square with x² having negative coefficient

Hai,

Thank you but would the x be equal to 2 ± 1 because of the square root b/c making a graph out of it doesn't seem to work out.

Last edited by Not Dav (2021-04-24 07:36:54)

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I wish thee not, yet it is always -Anonymous

mathland

Member

Registered: 2021-03-25

Posts: 444

Re: How to complete the square with x² having negative coefficient

Hai,

Thank you but would the x be equal to 2 ± 1 because of the square root b/c making a graph out of it doesn't seem to work out.

Yes, you are right. Whenever we take the square root, there is a positive and negative answer.

Look here:

x - 2 = 1

x = 1 + 2

x = 3

And

x - 2 = -1

x = -1 + 2

x = 1

Yep, that's it. I will now upgrade my earlier reply.

Last edited by mathland (2021-04-24 07:46:08)

mathland

Member

Registered: 2021-03-25

Posts: 444

Re: How to complete the square with x² having negative coefficient

Hai,

Thank you but would the x be equal to 2 ± 1 because of the square root b/c making a graph out of it doesn't seem to work out.

Tell me, which course are you taking? Algebra 2? College Algebra? Precalculus? Which one?

Not Dav

Member

From: Animal kingdom

Registered: 2021-04-22

Posts: 55

Re: How to complete the square with x² having negative coefficient

Algebra 2

Last edited by Not Dav (2021-04-24 08:35:10)

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Bob

Administrator

Registered: 2010-06-20

Posts: 10,191

Re: How to complete the square with x² having negative coefficient

Hi guys,

I've been doing my mother's garden today, so I've only just found time to check in.

All you have done is fine but here's some tricks that will save you time.

If b squared minus 4ac is a perfect square then the quadratic will factorise nicely.

In your example 4 times 4.  -4 times -1 times -3 = 4 so it's worth looking for a factorisation.

Y = (-x + 3)(+x -1) (by inspection) so y is 0 when x = 3 or 1

The vertex of any quadratic will always be midway between the roots so at x=2.

Hope that helps,

Bob

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mathland

Member

Registered: 2021-03-25

Posts: 444

Re: How to complete the square with x² having negative coefficient

Hi guys,

I've been doing my mother's garden today, so I've only just found time to check in.

All you have done is fine but here's some tricks that will save you time.
⁹
If b squared minus 4ac is a perfect square then the quadratic will factorise nicely.

In your example 4 times 4.  -4 times -1 times -3 = 4 so it's worth looking for a factorisation.

Y = (-x + 3)(+x -1) (by inspection) so y is 0 when x = 3 or 1

The vertex of any quadratic will always be midway between the roots so at x=2.

Hope that helps,

Bob

I know the quadratic formula works in this case but I did not want to show him a simple plug and chug. I want him to understand what is going on.

Last edited by mathland (2021-04-24 10:44:17)

mathland

Member

Registered: 2021-03-25

Posts: 444

Re: How to complete the square with x² having negative coefficient

Algebra 2

Post your algebra 2 questions here. I'm ok with algebra 2 topics.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,191

Re: How to complete the square with x² having negative coefficient

I know the quadratic formula works in this case but I did not want to show him a simple plug and chug. I want him to understand what is going on.

Me too.  Which is why I didn't use the quadratic formula.  Direct factorisation requires thought, not 'plug and chug'.

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

mathland

Member

Registered: 2021-03-25

Posts: 444

Re: How to complete the square with x² having negative coefficient

I know the quadratic formula works in this case but I did not want to show him a simple plug and chug. I want him to understand what is going on.

Me too.  Which is why I didn't use the quadratic formula.  Direct factorisation requires thought, not 'plug and chug'.

Bob

Sounds good. I like the site but sure wish more people would get involved.

zetafunc

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Posts: 2,432

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Re: How to complete the square with x² having negative coefficient

y = -x² +4x -3

y= (-x² +4x -3)/(-1)

y = x²  - 4x + 3

We can now complete the square.

Set equation to 0.

x²  - 4x + 3 = 0

x²  - 4x = -3

You need to set the equation equal to 0 before you divide the quadratic through by -1, not after: otherwise you're effectively saying that -x^2 + 4x - 3 and x^2 - 4x + 3 are equivalent, which they are not.

mathland

Member

Registered: 2021-03-25

Posts: 444

Re: How to complete the square with x² having negative coefficient

y = -x² +4x -3

y= (-x² +4x -3)/(-1)

y = x²  - 4x + 3

We can now complete the square.

Set equation to 0.

x²  - 4x + 3 = 0

x²  - 4x = -3

You need to set the equation equal to 0 before you divide the quadratic through by -1, not after: otherwise you're effectively saying that -x^2 + 4x - 3 and x^2 - 4x + 3 are equivalent, which they are not.

I got it.