Question

The first two steps in the derivation of the quadratic formula by completing the square are shown below. Which answer choice shows the correct next step?

A. ax+b=32
B. ax ² +bx+4=0
C. ax+b=−4
D. ax² +bx+32=0

Answer 1

The correct next step in completing the square to derive the quadratic formula is to add the square of half of the coefficient of x to both sides of the equation.

Step-by-step explanation:

The first two steps in the derivation of the quadratic formula by completing the square are:

1. Start with the quadratic equation in the form ax² + bx + c = 0.
2. Move the constant term c to the other side of the equation to isolate the x terms: ax² + bx = -c.
3. Add the square of half of the coefficient of x to both sides of the equation: ax² + bx + (b/2)² = -c + (b/2)². This step effectively completes the square on the left side of the equation.

The correct next step is D. ax² + bx + 32 = 0. This equation is obtained by replacing the constant term c with 32 on the right side of the equation, resulting in ax² + bx = -32.