If $y=a(x-2)^{2}+c$ and $y=(2 x-5)(x-b)$ represent the same quadratic function, the value of $b$ is
(A) 3
(B) $\frac{3}{2}$
(C) $\frac{4}{5}$
(D) $-\frac{5}{2}$
(E) $\frac{8}{5}$
Solution 1

We expand the two expressions.

First, $y=a(x-2)^{2}+c=a\left(x^{2}-4 x+4\right)+c=a x^{2}-4 a x+(4 a+c)$.

Second, $y=(2 x-5)(x-b)=2 x^{2}-(5+2 b) x+5 b$.

We can then equate the coefficients.

From the leading coefficients, $a=2$.

From the coefficients of $x, 4 a=8=5+2 b$ or $b=\frac{3}{2}$.

## Solution 2

The $x$-coordinate of the vertex of a parabola is the average value of the roots of the parabola. From the first of the given forms, the vertex has $x$ coordinate 2 .

From the second of the given forms, the roots are $x=\frac{5}{2}$ and $x=b$.

Therefore, $\frac{1}{2}\left(\frac{5}{2}+b\right)=2$ or $\frac{5}{2}+b=4$ or $b=\frac{3}{2}$.

Answer: (B)