Question

Complete the square to write the function given below in the form f(x)=a(x−h)^2+k. f(x)=−x2−10x−38

Answer 1

f(x) = -1(x+5)^2 - 13

Explanation:

f(x) = -x2 -10x -38

Start by noticing that a=-1, and factor that out from the x^2 and x term:

f(x) = -1(x^2+10x) - 38

Our b term is now 10. Find the new c such that c = (b/2)^2:

c = (10/2)^2 = 5^2 = 25

Add and subtract that from the inside:

f(x) = -1(x^2 + 10x + 25 - 25) -38

Take the -25, and move it to the outside. Because a = -1, we actually add it to the outside:

f(x) = -1(x^2 + 10x + 25) + 25 - 38

Simplify:

f(x) = -1(x^2 + 10x + 25) - 13

Because c = (b/2)^2, we can write everything in the parenthesis as a perfect square with h = (b/2) =
`√(c)`:

f(x) = -1(x+5)^2 - 13

And that gives us the final answer. This answer in vertex form gives us the vertex (-5, -13).