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Completing the square

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  • 1. 5-4 Completing the Square 5-4 Completing the Square Warm Up Lesson Presentation Lesson Quiz Holt Algebra Holt Algebra 2 2
  • 2. 5-4 Completing the Square Warm Up Write each expression as a trinomial. 1. (x – 5)2 2. (3x + 5)2 x2 – 10x + 25 9x2 + 30x + 25 Factor each expression. 3. x2 – 18 + 81 (x – 9)2 4. 16x2 + 24x + 9 (4x + 3)2 Holt Algebra 2
  • 3. 5-4 Completing the Square Objectives Solve quadratic equations by completing the square. Holt Algebra 2
  • 4. 5-4 Completing the Square Vocabulary completing the square Holt Algebra 2
  • 5. 5-4 Completing the Square Many quadratic equations contain expressions that cannot be easily factored. For equations containing these types of expressions, you can use square roots to find roots. Holt Algebra 2
  • 6. 5-4 Completing the Square Reading Math Read Holt Algebra 2 as “plus or minus square root of a.”
  • 7. 5-4 Completing the Square Example 1A: Solving Equations by Using the Square Root Property Solve the equation. 4x2 + 11 = 59 4x2 = 48 x2 = 12 Subtract 11 from both sides. Divide both sides by 4 to isolate the square term. Take the square root of both sides. Simplify. Holt Algebra 2
  • 8. 5-4 Completing the Square Example 1B: Solving Equations by Using the Square Root Property Solve the equation. x2 + 12x + 36 = 28 (x + 6)2 = 28 Factor the perfect square trinomial Take the square root of both sides. Subtract 6 from both sides. Simplify. Holt Algebra 2
  • 9. 5-4 Completing the Square Check It Out! Example 1a Solve the equation. 4x2 – 20 = 5 4x2 = 25 x2 = 25 4 Add 20 to both sides. Divide both sides by 4 to isolate the square term. Take the square root of both sides. Simplify. Holt Algebra 2
  • 10. 5-4 Completing the Square Check It Out! Example 1b Solve the equation. x2 + 8x + 16 = 49 (x + 4)2 = 49 Factor the perfect square trinomial. Take the square root of both sides. x = –4 ± 49 Subtract 4 from both sides. x = –11, 3 Holt Algebra 2 Simplify.
  • 11. 5-4 Completing the Square If a quadratic expression of the form x2 + bx cannot model a square, you can add a term to form a perfect square trinomial. This is called completing the square. Holt Algebra 2
  • 12. 5-4 Completing the Square Example 2A: Completing the Square Complete the square for the expression. Write the resulting expression as a binomial squared. x2 – 14x + Find . Add. x2 – 14x + 49 Factor. (x – 7)2 Check Find the square of the binomial. (x – 7)2 = (x – 7)(x – 7) = x2 – 14x + 49 Holt Algebra 2
  • 13. 5-4 Completing the Square Example 2B: Completing the Square Complete the square for the expression. Write the resulting expression as a binomial squared. x2 + 9x + Find Add. Factor. Holt Algebra 2 . Check Find the square of the binomial.
  • 14. 5-4 Completing the Square Check It Out! Example 2a Complete the square for the expression. Write the resulting expression as a binomial squared. x2 + 4x + Find x2 + 4x + 4 (x + 2)2 Check . Add. Factor. Find the square of the binomial. (x + 2)2 = (x + 2)(x + 2) = x2 + 4x + 4 Holt Algebra 2
  • 15. 5-4 Completing the Square Check It Out! Example 2b Complete the square for the expression. Write the resulting expression as a binomial squared. x2 – 4x + Find x2 – 4x + 4 (x – 2)2 Check Holt Algebra 2 . Add. Factor. Find the square of the binomial. (x – 2)2 = (x – 2)(x – 2) = x2 – 4x + 4
  • 16. 5-4 Completing the Square Check It Out! Example 2c Complete the square for the expression. Write the resulting expression as a binomial squared. x2 + 3x + Find Add. Factor. Holt Algebra 2 . Check Find the square of the binomial.
  • 17. 5-4 Completing the Square You can complete the square to solve quadratic equations. Holt Algebra 2
  • 18. 5-4 Completing the Square Example 3A: Solving a Quadratic Equation by Completing the Square Solve the equation by completing the square. x2 = 12x – 20 x – 12x = –20 Collect variable terms on one side. x2 – 12x + Set up to complete the square. 2 = –20 + Add x2 – 12x + 36 = –20 + 36 Holt Algebra 2 to both sides. Simplify.
  • 19. 5-4 Completing the Square Example 3A Continued (x – 6)2 = 16 Factor. Take the square root of both sides. x – 6 = ±4 x – 6 = 4 or x – 6 = –4 x = 10 or x = 2 Holt Algebra 2 Simplify. Solve for x.
  • 20. 5-4 Completing the Square Example 3B: Solving a Quadratic Equation by Completing the Square Solve the equation by completing the square. 18x + 3x2 = 45 x2 + 6x = 15 x2 + 6x + = 15 + Divide both sides by 3. Set up to complete the square. Add x2 + 6x + 9 = 15 + 9 Holt Algebra 2 to both sides. Simplify.
  • 21. 5-4 Completing the Square Example 3B Continued (x + 3)2 = 24 Factor. Take the square root of both sides. Simplify. Holt Algebra 2
  • 22. 5-4 Completing the Square Check It Out! Example 3a Solve the equation by completing the square. x2 – 2 = 9x Collect variable terms on one side. x2 – 9x = 2 x2 – 9x + =2+ Set up to complete the square. Add to both sides. Simplify. Holt Algebra 2
  • 23. 5-4 Completing the Square Check It Out! Example 3a Continued Factor. x– 9 = ± 89 4 2 x= Holt Algebra 2 9 ± 89 2 Take the square root of both sides. Simplify.
  • 24. 5-4 Completing the Square Check It Out! Example 3b Solve the equation by completing the square. 3x2 – 24x = 27 Divide both sides by 3. x2 – 8x = 9 x2 –8x + =9+ Set up to complete the square. Add to both sides. Simplify. Holt Algebra 2
  • 25. 5-4 Completing the Square Check It Out! Example 3b Continued Solve the equation by completing the square. Factor. Take the square root of both sides. Simplify. x – 4 =–5 or x – 4 = 5 x =–1 or x = 9 Holt Algebra 2 Solve for x.