Which equation is [tex]y = 2x^2 - 8x + 9[/tex] rewritten in vertex form?

A. [tex]y = 2(x - 2)^2 + 9[/tex]
B. [tex]y = 2(x - 2)^2 + 5[/tex]
C. [tex]y = 2(x - 2)^2 + 1[/tex]
D. [tex]y = 2(x - 2)^2 + 17[/tex]



Answer :

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]