How do you find the value of c that makes #x^2-24x+c# into a perfect square?

1 Answer
Jan 5, 2018

Answer:

Divide the coefficient of #x# (not #x^2#) by 2, and then square that.
#c=144#.

Explanation:

Divide the coefficient of #x# (not #x^2#) by 2, and then square that.

So for #x^2-24x#, dividing #-24# by #2# gets us #-12#.
Squaring #-12# gets us #144#,

So #c=144#.


In general:

Consider the factoring identity

#\qquad(x-a)^2 = x^2 - 2ax+ a^2#

If we only have the middle term, #- 2ax#, then dividing the middle term by #2x# gets us

#\qquad\frac{-2ax}{2x} = -a#

(If we only take the coefficient on #x#, we do not have to divide by #x#.)

Then if we square that, we get

#\qquad(-a)^2 = a^2#

So as you can see, taking the coefficient on #x#, dividing it by #2#, and then squaring it will get us the magic constant that lets us complete the square.