Completing the square.

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Sep 23rd 2010, 08:27 AM
greatersanta616

Completing the square.

A level Mathematics. Use the method of completing the square to solve these quadratic equations. Give your answer in the form a(+or-)b root "n" where a and b are rational, and n is an integer.
a. x^2+4x-1=0
d. x^2-8x-3=0

Please help, as I do not understand fully how to do this.
Sep 23rd 2010, 08:39 AM
Archie Meade

Quote:

Originally Posted by greatersanta616

A level Mathematics. Use the method of completing the square to solve these quadratic equations. Give your answer in the form a(+or-)b root "n" where a and b are rational, and n is an integer.
a. x^2+4x-1=0
d. x^2-8x-3=0

Please help, as I do not understand fully how to do this.

(a)

$(x+p)^2=(x+p)(x+p)=x(x+p)+p(x+p)=x^2+2xp+p^2$

Therefore, in $x^2+4x=1$ the "2p" term is 4, so p is 2.

But... $(x+2)^2=x^2+4x+4$

Hence

$x^2+4x=1\Rightarrow\ x^2+4x+4=1+4$

$(x+2)^2=5$

$x+2=\pm\sqrt{5}$

$x=-2\pm\sqrt{5}$

Try (d)
Sep 23rd 2010, 08:46 AM
greatersanta616

(d) x^2-8x-3=0
So..
(x-4)^2-19=0
Therefore.
(x-4)^2=19
henceforth...
x-4=(+or-)root 19
and...
x=4(+or-)root 19

Is this ok?
Sep 23rd 2010, 08:49 AM
Archie Meade

Quote:

Originally Posted by greatersanta616

(d) x^2-8x-3=0
So..
(x-4)^2-19=0
Therefore.
(x-4)^2=19
henceforth...
x-4=(+or-)root 19
and...
x=4(+or-)root 19

Is this ok?

Yes, that's the completing the square technique..you got it fast!

I think that "a" and "b" being rational, would apply more to the case of using the quadratic formula,
however, those values, while being integers, are rational anyway.
Sep 23rd 2010, 08:51 AM
greatersanta616

Quote:

Originally Posted by Archie Meade

Yes, that's the completing the square technique..you got it fast!

I think that "a" and "b" being rational, would apply more to the case of using the quadratic formula,
however, those values, while being integers, are rational anyway.

Thank you very much for this. It really helped.