There are 5 quadratics below. Four of them have two distinct roots each. The other has only one distinct root; find the value of that root.
\begin{align*} &4x^2 +16x - 9\\ &2x^2 + 80x + 400\\ &x^2 - 6x - 9\\ &4x^2 - 12x + 9\\ &{-x^2 + 14x + 49} \end{align*}
A quadratic will have only one root when the discriminant, b^2 - 4ac = 0
4x^2 + 16x - 9 ⇒ 16^2 - 4(4)(-9) > 0
2x^2 + 80x + 400 ⇒ 80^2 - 4(2 )( 400) > 0
x^2 - 6x - 9 ⇒ (-6)^2 - 4(1)(-9) > 0
4x^2 - 12x + 9 ⇒ (-12)^2 - 4 (4)(9) = 0
-x^2 + 14x + 49 ⇒ (14)^2 - 4(-1)(49) > 0
So
4x^2 - 12x + 9 = 0 factors as
(2x - 3) (2x - 3) = 0
(2x - 3)^2 = 0
And this is true when x = 3/2