find the vertex form of f(x)=2x^2+24x+70

1. Start with the given function:

f(x) = 2x^2 + 24x + 70

2. Factor out the coefficient of \(x^2\) from the first two terms:

f(x) = 2(x^2 + 12x) + 70

3. To complete the square, take the coefficient of \(x\), divide it by 2, and square it:

(12 / 2)^2 = 36

4. Add and subtract this square inside the brackets:

f(x) = 2(x^2 + 12x + 36 - 36) + 70

5. Group the perfect square trinomial and simplify:

f(x) = 2((x + 6)^2 - 36) + 70

6. Distribute the 2 and combine constants:

f(x) = 2(x + 6)^2 - 72 + 70

f(x) = 2(x + 6)^2 - 2

So the vertex form is:

f(x) = 2(x+6)^2 - 2