a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)² = q that has the same solutions. Derive the quadratic formula from this form.

This is a sneaky and fun way to solve unruly quadratic equations. It involves creating and factoring a "perfect square" (meaning, a quadratic expression whose two linear factors are the same). Let's solve the same equation as we did before: x² + x – 12 = 0.

First, add 12 to both sides: x² + x = 12. Next, complete the square by making the left side a perfect square. To do that, we take b, divide by 2, and square the result. In this case, our b = 1, which ends up being ¼. Add this to both sides of the equation.

The left side is a perfect square because its two linear factors are the same: x + ½. Therefore, we can re-write the left side of the equation and simplify the right-hand side.

From there, we can square root both sides and subtract both sides by ½.

x = 3, -4