Using the Completing the Square Method: Rewrite f(x) = x² + 10x + 7 in Vertex Form

To rewrite the quadratic function f(x) = x² + 10x + 7 in vertex form, we can use the completing the square method. The vertex form of a quadratic equation is given by:

f(x) = a(x – h)² + k

where (h, k) is the vertex of the parabola. Let’s follow these steps to complete the square:

1. Start with the given function:

f(x) = x² + 10x + 7

2. Focus on the x² and 10x terms. We want to complete the square for these terms:

x² + 10x

3. To complete the square, take half of the coefficient of x, which is 10, divide it by 2 to get 5, and then square it to get 25.
4. Add and subtract this square within the equation:

f(x) = (x² + 10x + 25) – 25 + 7

5. Now, simplify the equation:

f(x) = (x + 5)² – 18

Now we have the function written in vertex form:

f(x) = (x + 5)² – 18

The vertex of the parabola is (-5, -18).