How can I rewrite the quadratic function f(x) = x² + 10x – 7 in vertex form using the completing the square method?

To rewrite the quadratic function f(x) = x² + 10x – 7 in vertex form, we can follow the completing the square method.

Step 1: Identify the coefficients

We start with the equation in the standard form:

f(x) = x² + 10x - 7

Here, the coefficient of the x² term is 1, and the coefficient of the x term is 10.

Step 2: Group the x terms

Next, we focus on the x² + 10x portion of the equation:

f(x) = (x² + 10x) - 7

Step 3: Complete the square

To complete the square, we take half of the x coefficient, square it, and then add and subtract that value within the parentheses. Half of 10 is 5, and squaring it gives 25.

f(x) = (x² + 10x + 25 - 25) - 7

This can be simplified to:

f(x) = (x² + 10x + 25) - 25 - 7

Step 4: Rewrite as a perfect square

Now we can rewrite the quadratic expression as a perfect square:

f(x) = (x + 5)² - 32

Step 5: Final vertex form

The vertex form of the quadratic function is:

f(x) = (x + 5)² - 32

Thus, we see that the vertex of the parabola represented by the function f(x) = (x + 5)² – 32 is at the point (-5, -32).

Conclusion

To summarize, using the completing the square method, we have successfully transformed the original quadratic function into its vertex form, which is f(x) = (x + 5)² – 32.