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10-5: Solve Quadratic Equations by Completing the Square

10-5: Solve Quadratic Equations by Completing the Square. Objectives: Solve quadratic equations by completing the square Common Core Standards: A-SSE-3, A-REI-4 Assessments: Define all vocabulary from this section Do worksheet 10-5.

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10-5: Solve Quadratic Equations by Completing the Square

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  1. 10-5: Solve Quadratic Equations by Completing the Square Objectives: Solve quadratic equations by completing the square Common Core Standards: A-SSE-3, A-REI-4 Assessments: Define all vocabulary from this section Do worksheet 10-5

  2. In the previous lesson, you solved quadratic equations by isolating x2 and then using square roots. This method works if the quadratic equation, when written in standard form, is a perfect square. When a trinomial is a perfect square, there is a relationship between the coefficient of the x-term and the constant term. X2 + 6x + 9 x2– 8x + 16 Divide the coefficient of the x-term by 2, then square the result to get the constant term.

  3. An expression in the form x2 + bxis not aperfect square. However, you can use the relationship shown above to add a term to x2 + bx to form a trinomial that is a perfect square. This is called completing the square.

  4. A. x2 + 2x + B. x2 – 6x+ . Complete the square to form a perfect square trinomial. x2 + 2x x2 + –6x Identify b. x2 + 2x + 1 x2 – 6x + 9

  5. b 5 2 2 = = x x 2 2 + + 25 5 2 c = = 4 2 • Find the value of cthat makes the expression x2 + 5x + ca perfect square trinomial. • Then write the expression as the square of a binomial. STEP 1 Find the value ofc.For the expression to be a perfect squaretrinomial, c needs to be the square of half thecoefficient ofbx. Find the square of half the coefficient of bx. STEP 2 Write the expression as a perfect square trinomial. Then write the expression as the square of a binomial. Square of a binomial

  6. Find the value of cthat makes the expression a perfect square trinomial. • Then write the expression as the square of a binomial. 1. x2 + 8x + c 2. x2 12x + c 3. x2 + 3x + c

  7. 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. Solving a Quadratic Equation by Completing the Square

  8. . Step 2 Step 6x + 8 = 7 or x + 8 = –7 x = –1 or x = –15 Solve by completing the square. Check your answer. The equation is in the form x2 + bx = c. x2 + 16x = –15 Step 1 x2 + 16x = –15 Step 3x2 + 16x + 64 = –15 + 64 Complete the square. Step 4 (x + 8)2 = 49 Factor and simplify. Step 5 x + 8 = ± 7 Take the square root of both sides. Write and solve two equations.

  9. (x – 2)2= –1 Step 4 Solve by completing the square. –3x2 + 12x – 15 = 0 Factor and simplify. There is no real number whose square is negative, so there are no real solutions.

  10. Solve x2 – 16x = –15 by completing the square.

  11. Solve a quadratic equation in standard form Solve 2x2 + 20x – 8 = 0 by completing the square.

  12. 4. x2 – 2x = 3 5. m2 + 10m = –8 6.3g2 – 24g+ 27 = 0

  13. You decide to use chalkboard paint to create a chalkboard on a door. You want the chalkboard to have a uniform border as shown. You have enough chalkboard paint to cover 6 square feet. Find the width of the border to the nearest inch.

  14. (7 – 2x) (3 – 2x) 6 = Write a verbal model. Then write an equation. Let x be the width (in feet) of the border.

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