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9-3B Completing the Square Adapted from Math 8H Completing the Square JoAnn Evans (to solve quadratic equations) You will need a calculator for today’s lesson. Add a few problems where the middle term is odd!

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prime

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First, a few review topics to refresh your memory……… Solving a quadratic by taking the square root: To solve for x, take the square root of both sides of the equation. When taking the square root of both sides of the equation, you must add a plus/minus sign because 81 has both a positive and a negative root.

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Another look at the characteristics common to all PERFECT SQUARE TRINOMIALS : x 2 + 6x + 9 x 2 - 12x + 36 x 2 + 2x + 1 x 2 + x + ¼ In each trinomial, take a look at the coefficient of the middle term and the constant. What is the relationship between these two numbers? Take half of the middle term coefficient. If you square that number, you will have the constant in each case. Why is that? Think of the factored form of each P.S.T. (x + 3) 2 (x – 6) 2 (x + 1) 2 (x + ½) 2

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What is the missing constant that will complete the square?

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Make each incomplete trinomial into a PERFECT SQUARE TRINOMIAL by finding the missing constant: x 2 + 8x + ___ x 2 – 14x + ___ x 2 + 10x + ___ x 2 + 22x + ___ x 2 - 20x + ___ 16 49 25 121 100 Factored form: (x + 4) 2 Factored form: (x - 7) 2 Factored form: (x + 5) 2 Factored form: (x + 11) 2 Factored form: (x - 10) 2

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To solve a quadratic by “Completing the Square”, follow these steps: 1.Make sure the coefficient of x 2 is 1. 2. Move everything to the LEFT side of the equation EXCEPT the constant. 3. Make the left hand side of the equation into a PERFECT SQUARE TRINOMIAL by following the steps we just practiced. 4. Remember, if you add a number to one side of an equation, you must add the same number to the other side of the equation. 5. Factor the left side into the SQUARE OF A BINOMIAL. 6. Take the square root of each side. Remember to add the ± symbol to the right side. 7. Solve for x.

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(x – 3) 2 = 2 (half of -6) 2 Solve by completing the square. x 2 – 6x + 7 = 0 x 2 – 6x + ___ = -7 x 2 – 6x + 9 = -7 + 9 x 2 – 6x + 9 = 2 Simplify. Find the square root of each side. Solve for x. Add 9 to both sides. Keep everything except the constant on the left side. Leave a space for the number that will make a perfect square trinomial. What’s half of -6, squared? Rewrite as a binomial squared.

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Example 1 Solve by completing the square. Round to the nearest tenth if necessary. x 2 – 4x + 2 = 0 Keep everything except the constant on the left side. x 2 – 4x + ___ = -2 Leave a space for the number that will make a perfect square trinomial. What’s half of -4, squared? x 2 – 4x + 4 = -2 + 4 x 2 – 4x + 4 = 2 Simplify. (half of -4) 2 (x – 2) 2 = 2 Rewrite as a binomial squared. Find the square root of each side. Solve for x. Add 4 to both sides.

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Example 2 Solve by completing the square. Round to nearest tenth if necessary. x 2 - 12x + 4 = 0 Keep everything except the constant on the left side. x 2 - 12x + ___ = -4 Leave a space for the number that will make a perfect square trinomial. What’s half of -12, squared? x 2 - 12x + 36 = -4 + 36 x 2 - 12x + 36 = 32 Simplify. (half of -12) 2 (x - 6) 2 = 32 Rewrite as a binomial squared. Find the square root of each side. Solve for x. Add 36 to both sides.

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Solve by completing the square. Round to the nearest tenth if necessary. x 2 + 6x - 8 = 0 Example 3 Example 4 x 2 - 2x - 3 = 0 Example 5 x 2 + 16x + 4 = 0 Example 6 x 2 + 10x - 24 = 0

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Example 3 Solve by completing the square. Round to the nearest tenth if necessary x 2 + 6x - 8 = 0 Keep everything except the constant on the left side. x 2 + 6x + ___ = 8 Leave a space for the number that will make a perfect square trinomial. What’s half of 6, squared? x 2 + 6x + 9 = 8 + 9 x 2 + 6x + 9 = 17 Simplify. (half of 6) 2 (x + 3) 2 = 17 Rewrite as a binomial squared. Find the square root of each side. Solve for x. Add 9 to both sides.

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Example 4 Solve by completing the square. Round to the nearest tenth if necessary. x 2 - 2x - 3 = 0 Keep everything except the constant on the left side. x 2 - 2x + ___ = 3 Leave a space for the number that will make a perfect square trinomial. What’s half of -2, squared? x 2 - 2x + 1 = 3 + 1 x 2 - 2x + 1 = 4 Simplify. (half of -2) 2 (x - 1) 2 = 4 Rewrite as a binomial squared. Find the square root of each side. Solve for x. Add 1 to both sides.

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Example 5 Solve by completing the square. Round to the nearest tenth if necessary. x 2 + 16x + 4 = 0 Keep everything except the constant on the left side. x 2 + 16x + ___ = -4 Leave a space for the number that will make a perfect square trinomial. What’s half of 16, squared? x 2 + 16x + 64 = -4 + 64 x 2 + 16x + 64 = 60 Simplify. (half of 16) 2 (x + 8) 2 = 60 Rewrite as a binomial squared. Find the square root of each side. Solve for x. Add 64 to both sides.

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Example 6 Solve by completing the square. Round to the nearest tenth if necessary. x 2 + 10x - 24 = 0 Keep everything except the constant on the left side. x 2 + 10x + ___ = 24 Leave a space for the number that will make a perfect square trinomial. What’s half of 5, squared? x 2 + 10x + 25 = 24 + 25 x 2 + 10x + 25 = 49 Simplify. (half of 10) 2 (x + 5) 2 = 49 Rewrite as a binomial squared. Find the square root of each side. Solve for x. Add 25 to both sides.

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Solve by completing the square. x 2 + 6x - 8 = 0 Example 3 Example 4 x 2 - 2x - 3 = 0 Example 5 x 2 + 16x + 4 = 0 Example 6 x 2 + 10x - 24 = 0

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9-A9 Page 489-491, # 16–27, 48-52, 62-65.

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