Explain the process to complete the square in a quadratic equation.

To complete the square in a quadratic equation, add and subtract the square of half the coefficient of x.

Completing the square is a technique used to rewrite a quadratic equation in a standard form, which makes it easier to solve. The process involves adding and subtracting a constant term to both sides of the equation to create a perfect square trinomial.

To complete the square for an equation of the form ax^2 + bx + c = 0, first divide both sides by a to get x^2 + (b/a)x + (c/a) = 0. Then, take half of the coefficient of x, square it, and add and subtract that value to the right-hand side of the equation:

x^2 + (b/a)x + (b/2a)^2 - (b/2a)^2 + (c/a) = 0 + (b/2a)^2 - (c/a)

The left-hand side can be simplified to a perfect square trinomial:

(x + b/2a)^2 = (b/2a)^2 - (c/a)

Taking the square root of both sides and solving for x gives:

x = (-b ± √(b^2 - 4ac)) / 2a

This is the quadratic formula, which can be used to solve any quadratic equation. Completing the square is a useful technique for deriving the quadratic formula and for solving quadratic equations that cannot be factored easily.