What value makes c a perfect square 4x^2+12x+c?

Apr 8, 2015

I may perhaps be mistaking but i think the question should be: "For what value of $c$ will the expression $4 {x}^{2} + 12 x + c$ be a perfect square?"

In that case here's my solution:

That expression must be in the ${\left(a x + b\right)}^{2}$ for it to be a perfect square, so i write

$4 {x}^{2} + 12 x + c \equiv {\left(a x + b\right)}^{2}$

$\implies$4x^2 + 12x + c-= a^2x^2 + 2abx + b^2#

Equating coefficients of the powers of $x$ on both sides,

$4 = {a}^{2} \implies {a}^{2} = 4$

$12 = 2 a b \implies 4 {a}^{2} {b}^{2} = 144$$\implies 4 \cdot 4 \cdot {b}^{2} = 144 \implies {b}^{2} = 9$

$c = {b}^{2}$
$\implies c = 9$