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Introduction A trinomial of the form that can be written as the square of a binomial is called a perfect square trinomial. We can solve quadratic equations by transforming the left side of the equation into a perfect square trinomial and using square roots to solve. 1 5.2.3: Completing the Square

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Key Concepts When the binomial (x + a) is squared, the resulting perfect square trinomial is x 2 + 2ax + a 2. When the binomial (ax + b) is squared, the resulting perfect square trinomial is a 2 x 2 + 2abx + b 2. 2 5.2.3: Completing the Square

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Key Concepts, continued 3 5.2.3: Completing the Square Completing the Square to Solve Quadratics 1.Make sure the equation is in standard form, ax 2 + bx + c. 2.Subtract c from both sides. 3.Divide each term by a to get a leading coefficient of 1. 4.Add the square of half of the coefficient of the x-term to both sides to complete the square. 5.Express the perfect square trinomial as the square of a binomial. 6.Solve by using square roots.

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Common Errors/Misconceptions neglecting to add the c term of the perfect square trinomial to both sides not isolating x after squaring both sides forgetting that, when taking the square root, both the positive and negative roots must be considered (±) 4 5.2.3: Completing the Square

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Guided Practice Example 2 Solve x 2 + 6x + 4 = 0 by completing the square. 5 5.2.3: Completing the Square

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Guided Practice: Example 2, continued 1.Determine if x 2 + 6x + 4 is a perfect square trinomial. Take half of the value of b and then square the result. If this is equal to the value of c, then the expression is a perfect square trinomial. x 2 + 6x + 4 is not a perfect square trinomial because the square of half of 6 is not 4. 6 5.2.3: Completing the Square

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Guided Practice: Example 2, continued 2.Complete the square. 7 5.2.3: Completing the Square x 2 + 6x + 4 = 0Original equation x 2 + 6x = –4 Subtract 4 from both sides. x 2 + 6x + 3 2 = –4 + 3 2 Add the square of half of the coefficient of the x- term to both sides to complete the square. x 2 + 6x + 9 = 5Simplify.

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Guided Practice: Example 2, continued 3.Express the perfect square trinomial as the square of a binomial. Half of b is 3, so the left side of the equation can be written as (x + 3) 2. (x + 3) 2 = 5 8 5.2.3: Completing the Square

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Guided Practice: Example 2, continued 4.Isolate x. 9 5.2.3: Completing the Square (x + 3) 2 = 5Equation Take the square root of both sides. Subtract 3 from both sides.

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Guided Practice: Example 2, continued 5.Determine the solution(s). The equation x 2 + 6x + 4 = 0 has two solutions, 10 5.2.3: Completing the Square ✔

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Guided Practice: Example 2, continued 11 5.2.3: Completing the Square

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Guided Practice Example 3 Solve 5x 2 – 50x – 120 = 0 by completing the square. 12 5.2.3: Completing the Square

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Guided Practice: Example 3, continued 1.Determine if 5x 2 – 50x – 120 = 0 is a perfect square trinomial. The leading coefficient is not 1. First divide both sides of the equation by 5 so that a = 1. 13 5.2.3: Completing the Square 5x 2 – 50x – 120 = 0Original equation x 2 – 10x – 24 = 0Divide both sides by 5.

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Guided Practice: Example 3, continued Now that the leading coefficient is 1, take half of the value of b and then square the result. If the expression is equal to the value of c, then it is a perfect square trinomial. 5x 2 – 50x – 120 = 0 is not a perfect square trinomial because the square of half of –10 is not –24. 14 5.2.3: Completing the Square

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Guided Practice: Example 3, continued 2.Complete the square. 15 5.2.3: Completing the Square x 2 – 10x – 24 = 0Equation x 2 – 10x = 24Add 24 to both sides. x 2 – 10x + (–5) 2 = 24 + (–5) 2 Add the square of half of the coefficient of the x-term to both sides to complete the square. x 2 – 10x + 25 = 49Simplify.

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Guided Practice: Example 3, continued 3.Express the perfect square trinomial as the square of a binomial. Half of b is –5, so the left side of the equation can be written as (x – 5) 2. (x – 5) 2 = 49 16 5.2.3: Completing the Square

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Guided Practice: Example 3, continued 4.Isolate x. 17 5.2.3: Completing the Square (x – 5) 2 = 49Equation Take the square root of both sides. Add 5 to both sides. x = 5 + 7 = 12 or x = 5 – 7 = –2 Split the answer into two separate equations and solve for x.

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Guided Practice: Example 3, continued 5.Determine the solution(s). The equation 5x 2 – 50x – 120 = 0 has two solutions, x = –2 or x = 12. 18 5.2.3: Completing the Square ✔

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Guided Practice: Example 3, continued 19 5.2.3: Completing the Square

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