Solve x 2 + 14x + 49 = 64 by using the Square Root Property. Equation with Rational Roots Solve x 2 + 14x + 49 = 64 by using the Square Root Property. Original equation Factor the perfect square trinomial. Square Root Property Subtract 7 from each side. Example 1

x = –7 + 8 or x = –7 – 8 Write as two equations. Equation with Rational Roots x = –7 + 8 or x = –7 – 8 Write as two equations. x = 1 x = –15 Solve each equation. Answer: The solution set is {–15, 1}. Check: Substitute both values into the original equation. x 2 + 14x + 49 = 64 x 2 + 14x + 49 = 64 ? 1 2 + 14(1) + 49 = 64 (–15) 2 + 14(–15) + 49 = 64 ? 1 + 14 + 49 = 64 225 + (–210) + 49 = 64 64 = 64 64 = 64 Example 1

Solve x 2 – 4x + 4 = 13 by using the Square Root Property. Equation with Irrational Roots Solve x 2 – 4x + 4 = 13 by using the Square Root Property. Original equation Factor the perfect square trinomial. Square Root Property Add 2 to each side. Write as two equations. Use a calculator. Example 2

x 2 – 4x + 4 = 13 Original equation Equation with Irrational Roots Answer: The exact solutions of this equation are The approximate solutions are 5.61 and –1.61. Check these results by finding and graphing the related quadratic function. x 2 – 4x + 4 = 13 Original equation x 2 – 4x – 9 = 0 Subtract 13 from each side. y = x 2 – 4x – 9 Related quadratic function Example 2

Equation with Irrational Roots Check Use the ZERO function of a graphing calculator. The approximate zeros of the related function are –1.61 and 5.61. Example 2

A B C D Solve x 2 – 4x + 4 = 8 by using the Square Root Property. A. Example 2

Concept

Step 2 Square the result of Step 1. 62 = 36 Complete the Square Find the value of c that makes x 2 + 12x + c a perfect square. Then write the trinomial as a perfect square. Step 1 Find one half of 12. Step 2 Square the result of Step 1. 62 = 36 Step 3 Add the result of Step 2 to x 2 + 12x + 36 x 2 + 12x. Answer: The trinomial x2 + 12x + 36 can be written as (x + 6)2. Example 3

Find the value of c that makes x2 + 6x + c a perfect square Find the value of c that makes x2 + 6x + c a perfect square. Then write the trinomial as a perfect square. A. 9; (x + 3)2 B. 36; (x + 6)2 C. 9; (x – 3)2 D. 36; (x – 6)2 A B C D Example 3

Solve x2 + 4x – 12 = 0 by completing the square. Solve an Equation by Completing the Square Solve x2 + 4x – 12 = 0 by completing the square. x2 + 4x – 12 = 0 Notice that x2 + 4x – 12 is not a perfect square. x2 + 4x = 12 Rewrite so the left side is of the form x2 + bx. x2 + 4x + 4 = 12 + 4 add 4 to each side. (x + 2)2 = 16 Write the left side as a perfect square by factoring. Example 4

x + 2 = ± 4 Square Root Property Solve an Equation by Completing the Square x + 2 = ± 4 Square Root Property x = – 2 ± 4 Subtract 2 from each side. x = –2 + 4 or x = –2 – 4 Write as two equations. x = 2 x = –6 Solve each equation. Answer: The solution set is {–6, 2}. Example 4

Solve 3x2 – 2x – 1 = 0 by completing the square. Equation with a ≠ 1 Solve 3x2 – 2x – 1 = 0 by completing the square. 3x2 – 2x – 1 = 0 Notice that 3x2 – 2x – 1 is not a perfect square. Divide by the coefficient of the quadratic term, 3. Add to each side. Example 5

Equation with a ≠ 1 Write the left side as a perfect square by factoring. Simplify the right side. Square Root Property Example 5

or Write as two equations. Equation with a ≠ 1 or Write as two equations. x = 1 Solve each equation. Answer: Example 5

A B C D Solve 2x2 + 11x + 15 = 0 by completing the square. A. B. C. D. Example 5

Solve x 2 + 4x + 11 = 0 by completing the square. Equation with Imaginary Solutions Solve x 2 + 4x + 11 = 0 by completing the square. Notice that x 2 + 4x + 11 is not a perfect square. Rewrite so the left side is of the form x 2 + bx. Since , add 4 to each side. Write the left side as a perfect square. Square Root Property Example 6

Subtract 2 from each side. Equation with Imaginary Solutions Subtract 2 from each side. Example 6

End of the Lesson