The Square Root Property and Completing the Square Section 10.1 The Square Root Property and Completing the Square

Objectives Use the square root property to solve quadratic equations Solve quadratic equations by completing the square

Objective 1: Use the Square Root Property to Solve Quadratic Equations A quadratic equation is an equation of the form ax2 + bx + c = 0, where a, b, and c are real numbers and a ≠ 0. We have solved quadratic equations using factoring and the zero-factor property. The Square Root Property: For any non-negative real number c, if x2 = c, then:

EXAMPLE 1 Solve: x2 – 12 = 0 Strategy Since x2 – 12 does not factor as a difference of two integer squares, we must take an alternate approach. We will add 12 to both sides of the equation and use the square root property to solve for x. Why After adding 12 to both sides, the resulting equivalent equation will have the desired form x2 = c.

EXAMPLE 1 Solution Solve: x2 – 12 = 0 This is the equation to solve. It is a quadratic equation that is missing an x-term. To isolate x2 on the left side, add 12 to both sides. Use the square root property.

Objective 2: Solve Quadratic Equations by Completing the Square When the polynomial in a quadratic equation doesn’t factor easily, we can solve the equation by completing the square. This method is based on the following perfect-square trinomials (with leading coefficients of 1) and their factored forms: To complete the square on x2 + bx, add the square of one-half of the coefficient of x:

Objective 2: Solve Quadratic Equations by Completing the Square To solve an equation of the form ax2 + bx + c = 0 by completing the square, we use the following steps: If the coefficient of x2 is 1, go to step 2. If it is not, make it 1 by dividing both sides of the equation by the coefficient of x2. Get all variable terms on one side of the equation and constants on the other side. Complete the square by finding one-half of the coefficient of x, squaring the result, and adding the square to both sides of the equation. Factor the perfect-square trinomial as the square of a binomial. Solve the resulting equation using the square root property. Check your answers in the original equation.

EXAMPLE 6 Complete the square and factor the resulting perfect-square trinomial: Strategy We will add the square of one-half of the coefficient of x to the given binomial. Why Adding such a term will change the binomial into a perfect-square trinomial that will factor.

EXAMPLE 6 Complete the square and factor the resulting perfect-square trinomial: Solution a. To make x2 + 10x a perfect-square trinomial, we find one-half of 10, square it, and add the result to x2 + 10x. This trinomial factors as (x2 + 5)2. b. To make x2 – 11x a perfect-square trinomial, we find one-half of –11, square it, and add the result to x2 – 11x. This trinomial factors as .

EXAMPLE 6 Complete the square and factor the resulting perfect-square trinomial: Solution c. To make a perfect-square trinomial, we find one-half of , square it, and add the result to . This trinomial factors as .