Sure, let's proceed step-by-step to convert the quadratic equation [tex]\( y = 8x^2 + 32x + 17 \)[/tex] into vertex form.
1. Factor out the leading coefficient:
[tex]\[
y = 8(x^2 + 4x) + 17
\][/tex]
2. Form a perfect-square trinomial:
To complete the square inside the parenthesis, we need to add and subtract the square of half the coefficient of [tex]\( x \)[/tex]. The coefficient of [tex]\( x \)[/tex] in [tex]\( x^2 + 4x \)[/tex] is 4. Half of 4 is 2, and squared, it is [tex]\( 2^2 = 4 \)[/tex].
Add and subtract this square inside the parenthesis:
[tex]\[
y = 8\left(x^2 + 4x + 4 - 4\right) + 17
\][/tex]
Simplify:
[tex]\[
y = 8\left((x^2 + 4x + 4) - 4\right) + 17
\][/tex]
Factor the trinomial into a perfect square and distribute the 8:
[tex]\[
y = 8(x + 2)^2 - 8(4) + 17
\][/tex]
Which simplifies to:
[tex]\[
y = 8(x + 2)^2 - 32 + 17
\][/tex]
Therefore, we get:
[tex]\[
y = 8(x + 2)^2 - 15
\][/tex]
Now the quadratic equation [tex]\( y = 8x^2 + 32x + 17 \)[/tex] is converted to the vertex form [tex]\( y = 8(x + 2)^2 - 15 \)[/tex].
So, completing the statements:
[tex]\[
y = 8 \left( x^2 + 4x + 4 \right) - 32 + 17
\][/tex]
And the perfect-square trinomial itself is [tex]\( 8(x + 2)^2 \)[/tex].
