The equation of a parabola with roots [tex]\( x = -4 \)[/tex] and [tex]\( x = 2 \)[/tex] can be written in the factored form [tex]\( y = a(x + 4)(x - 2) \)[/tex]. We need to determine the value of [tex]\( a \)[/tex].
Given the point [tex]\((6, 10)\)[/tex] lies on the parabola, we substitute [tex]\( x = 6 \)[/tex] and [tex]\( y = 10 \)[/tex] into the equation [tex]\( y = a(x + 4)(x - 2) \)[/tex] and solve for [tex]\( a \)[/tex].
Let's substitute [tex]\( x = 6 \)[/tex] and [tex]\( y = 10 \)[/tex]:
[tex]\[ 10 = a(6 + 4)(6 - 2) \][/tex]
Simplifying inside the parentheses:
[tex]\[ 10 = a(10)(4) \][/tex]
[tex]\[ 10 = 40a \][/tex]
To solve for [tex]\( a \)[/tex]:
[tex]\[ a = \frac{10}{40} \][/tex]
[tex]\[ a = \frac{1}{4} \][/tex]
This is the correct equation that determines the value of [tex]\( a \)[/tex]:
[tex]\[ 10 = a(6 - 4)(6 + 2) \][/tex]
Thus, the correct answer is:
[tex]\[ 10 = a(6 - 4)(6 + 2) \][/tex]
