To rewrite the quadratic equation [tex]\( y = 2x^2 - 8x + 9 \)[/tex] in vertex form, we complete the square.
1. Start with the given equation:
[tex]\[
y = 2x^2 - 8x + 9
\][/tex]
2. Factor out the coefficient of [tex]\( x^2 \)[/tex] from the first two terms:
[tex]\[
y = 2(x^2 - 4x) + 9
\][/tex]
3. To complete the square inside the parentheses, take half the coefficient of [tex]\( x \)[/tex], square it, and add and subtract this square inside the parentheses:
[tex]\[
y = 2\left(x^2 - 4x + (-4/2)^2 - (-4/2)^2 \right) + 9
\][/tex]
4. Simplify inside the parentheses:
[tex]\[
y = 2\left(x^2 - 4x + 4 - 4 \right) + 9
\][/tex]
[tex]\[
y = 2\left((x - 2)^2 - 4 \right) + 9
\][/tex]
5. Distribute the 2 and combine like terms:
[tex]\[
y = 2(x - 2)^2 - 8 + 9
\][/tex]
[tex]\[
y = 2(x - 2)^2 + 1
\][/tex]
Hence, the equation in vertex form is:
[tex]\[
y = 2(x - 2)^2 + 1
\][/tex]
So the correct choice is:
[tex]\[
y = 2(x-2)^2 + 1
\][/tex]
