4 ObjectiveSolve quadratic equations by completing the square.
5 An expression in the form x2 + bx is not a perfect square An expression in the form x2 + bx is not a perfect square. However, you can use the algorithm below to add a term to x2 + bx to form a trinomial that is a perfect square. This is called completing the square.
6 Example 1: Completing the Square Complete the square to form a perfect square trinomial.A. x2 + 2x +B. x2 – 6x +x2 + 2xx2 + –6xIdentify b..x2 + 2x + 1x2 – 6x + 9
7 Check It Out! Example 1Complete the square to form a perfect square trinomial.a. x2 + 12x +b. x2 – 5x +x2 + 12xx2 + –5xIdentify b..x2 – 6x +x2 + 12x + 36
8 Check It Out! Example 1Complete the square to form a perfect square trinomial.c. 8x + x2 +x2 + 8xIdentify b..x2 + 12x + 16
9 To solve a quadratic equation in the form x2 + bx = c, first complete the square of x2 + bx. Then you can solve using square roots.
10 Solving a Quadratic Equation by Completing the Square
11 Example 2A: Solving x2 +bx = c Solve by completing the square.x2 + 16x = –15The equation is in the form x2 + bx = c.Step 1 x2 + 16x = –15Step 2.Step 3 x2 + 16x + 64 = –Complete the square.Step 4 (x + 8)2 = 49Factor and simplify.Take the square root of both sides.Step 5 x + 8 = ± 7Step 6 x + 8 = 7 or x + 8 = –7x = –1 or x = –15Write and solve two equations.
12 Example 2B: Solving x2 +bx = c Solve by completing the square.x2 – 4x – 6 = 0Write in the form x2 + bx = c.Step 1 x2 + (–4x) = 6Step 2.Step 3 x2 – 4x + 4 = 6 + 4Complete the square.Step 4 (x – 2)2 = 10Factor and simplify.Take the square root of both sides.Step 5 x – 2 = ± √10Step 6 x – 2 = √10 or x – 2 = –√10x = 2 + √10 or x = 2 – √10Write and solve two equations.
13 Check It Out! Example 2aSolve by completing the square.x2 + 10x = –9The equation is in the form x2 + bx = c.Step 1 x2 + 10x = –9Step 2.Step 3 x2 + 10x + 25 = –9 + 25Complete the square.Factor and simplify.Step 4 (x + 5)2 = 16Take the square root of both sides.Step 5 x + 5 = ± 4Step 6 x + 5 = 4 or x + 5 = –4x = –1 or x = –9Write and solve two equations.
14 Example 3A: Solving ax2 + bx = c by Completing the Square Solve by completing the square.–3x2 + 12x – 15 = 0Step 1Divide by – 3 to make a = 1.x2 – 4x + 5 = 0x2 – 4x = –5Write in the form x2 + bx = c.x2 + (–4x) = –5Step 2.Step 3x2 – 4x + 4 = –5 + 4Complete the square.
15 Example 3A ContinuedSolve by completing the square.Step 3x2 – 4x + 4 = –5 + 4–3x2 + 12x – 15 = 0Step 4(x – 2)2 = –1Factor and simplify.There is no real number whose square is negative, so there are no real solutions.
16 Lesson Quiz: Part IComplete the square to form a perfect square trinomial.1. x2 +11x +2. x2 – 18x +Solve by completing the square.3. x2 – 2x – 1 = 04. 3x2 + 6x = 1445. 4x x = 23816, –8
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