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Copyright © Cengage Learning. All rights reserved. Quadratic Equations, Quadratic Functions, and Complex Numbers 9

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Copyright © Cengage Learning. All rights reserved. Section 9.2 Solving Quadratic Equations by Completing the Square

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3 Objectives 1. Complete the square of a binomial to create a perfect trinomial square. 2. Solve a quadratic equation by completing the square. 1 1 2 2

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4 Solving Quadratic Equations by Completing the Square When the polynomial in a quadratic equation does not factor easily, we can solve the equation by using a method called completing the square. In fact, we can solve any quadratic equation by completing the square.

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5 Complete the square of a binomial to create a perfect trinomial square 1.

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6 Complete the square of a binomial to create a perfect trinomial square The method of completing the square is based on the following special products: x 2 + 2bx + b 2 = (x + b) 2 and x 2 – 2bx + b 2 = (x – b) 2 Recall that the trinomials x 2 + 2bx + b 2 and x 2 – 2bx + b 2 are both perfect trinomial squares, because each one factors as the square of a binomial. In each trinomial, if we take one-half of the coefficient of x and square the result, we get the third term. and

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7 To form a perfect trinomial square from the binomial x 2 + 12x, we take one-half of the coefficient x of (the 12), square the result, and add it to x 2 + 12x. This result is a perfect trinomial square, because x 2 + 12x + 36 = (x + 6) 2. Complete the square of a binomial to create a perfect trinomial square

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8 Example Form perfect trinomial squares using a. x 2 + 4x b. x 2 – 6x c. x 2 – 5x Solution: a. b. This is (x + 2) 2. This is (x – 3) 2.

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9 Example – Solution c. In each case, note that of the coefficient of x is the second term of the binomial factorization. cont’d

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10 Solve a quadratic equation by completing the square 2.

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11 Solve a quadratic equation by completing the square If the quadratic equation ax 2 + bx + c = 0 has a leading coefficient of 1 (a = 1) and especially if the middle term is even, we can solve by completing the square fairly easily.

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12 Exampl Solve by completing the square: x 2 – 8x – 20 = 0 Solution: We can solve the equation by completing the square. x 2 – 8x – 20 = 0 x 2 – 8x = 20 We then find one-half of the coefficient of x, square the result, and add it to both sides to make the left side a trinomial square. Add 20 to both sides.

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13 Example – Solution x 2 – 8x + 16 = 20 + 16 (x – 4) 2 = 36 x – 4 = x = 4 6 Simplify. Factor x 2 – 8x + 16 and simplify. Use the square root property to solve for x – 4. Add 4 to both sides, cont’d

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14 Example – Solution Because of the sign, there are two solutions. x = 4 + 6 or x = 4 – 6 x = 10 = –2 Check each solution. Note that this example could be solved by factoring. cont’d

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15 To solve a quadratic equation by completing the square, we follow these steps. Completing the Square 1. If necessary, write the quadratic equation in quadratic form, ax 2 + bx + c = 0. 2. If the coefficient of x 2 is not 1, divide both sides of the equation by a, the coefficient of x 2. 3. If necessary, add a number to both sides of the equation to place the constant term on the right side of the equal sign. Solve a quadratic equation by completing the square

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16 4. Complete the square: a. Find one-half of the coefficient of x and square it. b. Add the square to both sides of the equation. 5. Factor the perfect trinomial square on the left side of the equation and combine any like terms on the right side of the equation. 6. Use the square-root property to solve the resulting quadratic equation. 7. Check each solution in the original equation. Solve a quadratic equation by completing the square

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17 Example Solve by completing the square: 4x 2 –3 = –4x Solution: We first write the equation in quadratic form 4x 2 + 4x – 3 = 0 and then divide every term on both sides of the equation by 4 so that the coefficient of x 2 is 1. Add 4x to both sides.

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18 Example – Solution We then use completing the square to solve the equation. Simplify. Divide both sides by 4. Add to both sides. Add to both sides to complete the square. cont’d

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19 Example – Solution Check each solution. Note that this equation also can be solved by factoring. cont’d Factor and simplify. Solve for. Add to both sides. or

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