Complete the square to re-write the quadratic function in vertex form:

Question

Complete the square to re-write the quadratic function in vertex form:
y, equals, minus, 2, x, squared, plus, 12, x, minus, 24
y=−2x 
2
 +12x−24

GPT-4o mini · Answer

To rewrite the quadratic function $ y = -2x^2 + 12x - 24 $ in vertex form by completing the square, we can follow these steps: 1. **Factor out the coefficient of the $ x^2 $ term** (which is $-2$) from the first two terms: $$ y = -2(x^2 - 6x) - 24 $$ 2. **Complete the square inside the parentheses**. To do this, take the coefficient of $ x $ (which is $-6$), halve it (which gives $-3$), and then square it (which gives $9$): $$ y = -2(x^2 - 6x + 9 - 9) - 24 $$ Rearranging gives: $$ y = -2((x - 3)^2 - 9) - 24 $$ 3. **Distribute the $-2$**: $$ y = -2(x - 3)^2 + 18 - 24 $$ 4. **Simplify the constant terms**: $$ y = -2(x - 3)^2 - 6 $$ Now, the quadratic function in vertex form is: $$ y = -2(x - 3)^2 - 6 $$ The vertex of the function is at $(3, -6)$.