Completing the Square

[Guest]

What are the values of c that will make x^2+cx+49 a perfect square?

[Math Tutor]

c is 14
the formula is: [tex]x^2+cx+49 = x^2 + 2.7.x + 7^2 = (x + 7)^2[/tex]
Look at this: Polynomial Identities

[Guest]

[tex]c[/tex] could also be [tex]-14[/tex]

A perfect square is of the form [tex](ax+b)^2[/tex] which expands to give [tex]a^2 x^2 +2abx +b^2[/tex]
Comparing the coefficients with [tex]x^2+cx+49[/tex]
we get that
[tex]a^2 = 1[/tex],
[tex]2ab = c[/tex], and
[tex]b^2 = 49[/tex].
The first and last equations tell us that [tex]a= \pm 1[/tex], and [tex]b= \pm 7[/tex], so [tex]c=2ab = \pm14[/tex]. This just shows that the only possible values [tex]c[/tex] can take are [tex]14[/tex] and [tex]-14[/tex], we still need to show that there are perfect squares which achieve this, but this is easy enough to do, simply consider the expansions of [tex](x+7)^2[/tex] and [tex](x-7)^2[/tex].

Hope this helped,

R. Baber.

[Math Tutor]

Yeah you are right Mr. Baber.