To rewrite the quadratic equation [tex]\( y = -4x^2 + 2x - 7 \)[/tex] in vertex form [tex]\( y = a(x - h)^2 + k \)[/tex], we need to complete the square. The first step in this process is to factor out the coefficient of [tex]\( x^2 \)[/tex] from the terms involving [tex]\( x \)[/tex].
Here’s a detailed step-by-step explanation:
1. Factor out the coefficient of [tex]\( x^2 \)[/tex]:
The main objective is to isolate the quadratic part involving [tex]\( x \)[/tex] by factoring out the coefficient from both quadratic and linear terms. For the quadratic [tex]\( -4x^2 + 2x \)[/tex], the coefficient of [tex]\( x^2 \)[/tex] is [tex]\( -4 \)[/tex].
[tex]\[
y = -4(x^2 - \frac{1}{2}x) - 7
\][/tex]
This is the first step in rewriting the given quadratic function in vertex form [tex]\( y = a(x - h)^2 + k \)[/tex]. The next steps would typically involve completing the square within the parentheses and then adjusting the equation accordingly, but as per the given direction, the first step is to factor out the [tex]\( -4 \)[/tex] from the quadratic and linear terms.
