Five-Minute Check (over Lesson 3–4) Mathematical Practices Then/Now New Vocabulary Example 1: Equation with Rational Roots Example 2: Equation with Irrational Roots Key Concept: Completing the Square Example 3: Complete the Square Example 4: Solve an Equation by Completing the Square Example 5: Equation with a ≠ 1 Example 6: Equation with Imaginary Solutions Lesson Menu

Solve x 2 – x = 2 by factoring. A. x = 2 and x = –1 B. x = 1 and x = 2 C. x = 1 D. x = –1 and x = 1 5-Minute Check 1

Solve c 2 – 16c + 64 = 0 by factoring. A. x = 2 and x = –1 B. x = 1 and x = 2 C. x = 1 D. x = –1 and x = 1 5-Minute Check 2

Solve z 2 = 16z by factoring. A. z = 1 and z = 4 B. z = 0 and z = 16 C. z = –1 and z = 4 D. z = –16 5-Minute Check 3

Solve 2x 2 + 5x + 3 = 0 by factoring. A. x = and x = 2 B. x = 0 C. x = –1 D. x = and x = –1 5-Minute Check 4

Write a quadratic equation with the roots –1 and 6 in the form ax 2 + bx + c, where a, b, and c are integers. A. x2 – x + 6 = 0 B. x2 + x + 6 = 0 C. x2 – 5x – 6 = 0 D. x2 – 6x + 1 = 0 5-Minute Check 5

In a rectangle, the length is three inches greater than the width In a rectangle, the length is three inches greater than the width. The area of the rectangle is 108 square inches. Find the width of the rectangle. A. 6 in. B. 8 in. C. 9 in. D. 12 in. 5-Minute Check 6

A. 5 B. C. D. 5-Minute Check 1

A. B. C. D. 5-Minute Check 2

Simplify (5 + 7i) – (–3 + 2i). A. 2 + 9i B. 8 + 5i C. 2 – 9i D. –8 – 5i 5-Minute Check 3

Solve 7x2 + 63 = 0. A. ± 5i B. ± 3i C. ± 3 D. ± 3i – 3 5-Minute Check 4

What are the values of x and y when (4 + 2i) – (x + yi) = (2 + 5i)? A. x = 6, y = –7 B. x = –6, y = 7 C. x = –2, y = 3 D. x = 2, y = –3 5-Minute Check 5

Mathematical Practices 7 Look for and make use of structure. Content Standards N.CN.7 Solve quadratic equations with real coefficients that have complex solutions. F.IF.8.a Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. MP

You factored perfect square trinomials. Solve quadratic equations by using the Square Root Property. Solve quadratic equations by completing the square. Then/Now

completing the square Vocabulary

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

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 – 16x + 64 = 25 by using the Square Root Property. Equation with Rational Roots Solve x 2 – 16x + 64 = 25 by using the Square Root Property. x = –1 and x = 9 x = 11 and x = 21 C. x = 3 and x = 13 D. x = –13 and x = –3 Example 1

Solve x 2 – 4x + 4 = 13 by using the Square Root Property. Equation with Rational 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 Rational 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 Rational 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

Solve x 2 – 4x + 4 = 8 by using the Square Root Property. A. x = B. x = C. x = D. x = 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. c = 9; (x + 3)2 B. c = 36; (x + 6)2 C. c = 9; (x – 3)2 D. c = 36; (x – 6)2 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 is x = –6 and x = 2. Example 4

Solve x2 + 6x + 8 = 0 by completing the square. A. x = 2 and x = 4 B. x = -4 and x = -2 C. x = and x = D. x = 2 and x = -3 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

Solve 2x2 + 11x + 15 = 0 by completing the square. 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

Solve x 2 + 4x + 5 = 0 by completing the square. D. Example 6