Question

A quadratic function is defined by the expression x² − 14 x 42 . rewrite the expression in vertex form.

Answer 1

The quadratic expression x² − 14x + 42 can be rewritten in vertex form by completing the square, resulting in (x − 7)² − 7.

Step-by-step explanation:

The quadratic function given is x² − 14x + 42. To rewrite this expression in vertex form, we need to complete the square. The vertex form of a quadratic function is y = a(x − h)² + k, where (h, k) is the vertex of the parabola. Here's how you can transform it:

First, we need to factor a common factor of the quadratic and linear coefficients, but in this case, the quadratic coefficient is 1, so it is already in the simplest form. Thus, we proceed to the next steps:

1. Divide the linear coefficient (-14) by 2, getting -7, and then square it to obtain 49.
2. Add and subtract this number inside the quadratic expression to complete the square: x² − 14x + 49 - 49 + 42.
3. Rewrite the quadratic and the added square as a binomial square and simplify: (x − 7)² − 7 − 42.
4. Simplify the constants to get the complete vertex form: (x − 7)² − 7.

The expression in vertex form is therefore (x − 7)² − 7.