To find the x-coordinate of the vertex [tex]\( h \)[/tex] of a quadratic equation given in the form [tex]\( y = a(x + b)(x + c) \)[/tex], we can follow these steps:
1. Identify the Roots of the Quadratic Equation:
The given equation is [tex]\( y = a(x + 3)(x - 2) \)[/tex]. The roots (or zeros) of the equation occur where [tex]\( y = 0 \)[/tex]. This happens when each factor is zero.
[tex]\[
x + 3 = 0 \implies x = -3
\][/tex]
[tex]\[
x - 2 = 0 \implies x = 2
\][/tex]
2. Calculate the Midpoint of the Roots:
The x-coordinate of the vertex [tex]\( h \)[/tex] is the midpoint of the roots [tex]\( -3 \)[/tex] and [tex]\( 2 \)[/tex]. The midpoint formula for two points [tex]\( x_1 \)[/tex] and [tex]\( x_2 \)[/tex] is:
[tex]\[
h = \frac{x_1 + x_2}{2}
\][/tex]
Substituting the roots [tex]\( x_1 = -3 \)[/tex] and [tex]\( x_2 = 2 \)[/tex]:
[tex]\[
h = \frac{-3 + 2}{2}
\][/tex]
3. Simplify the Expression:
Simplifying the fraction:
[tex]\[
h = \frac{-3 + 2}{2} = \frac{-1}{2} = -0.5
\][/tex]
Thus, the x-coordinate of the vertex [tex]\( h \)[/tex] is [tex]\( -0.5 \)[/tex]. Therefore, the answer is:
[tex]\[
(A) \ - \frac{1}{2}
\][/tex]
