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Mastering Quadratic Equations: Completing the Square Techniques

In this chapter, we explore the method of completing the square to solve quadratic equations effectively. This technique is especially useful for equations that are difficult to factor. We delve into the square root property and provide step-by-step examples for solving specific equations. Through guided practice, students will learn to rewrite quadratic functions in vertex form. By the end of this lesson, students will have a firm grasp of completing the square, laying a foundation for future topics, including complex numbers and roots.

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Mastering Quadratic Equations: Completing the Square Techniques

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  1. Chapter 2 2-4 completing the square

  2. objectives *Solve quadratic equations by using the square property • * Solve quadratic equations by completing the square.

  3. Square root property • 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.

  4. Examples using the square root property Example #1 Solve the equation. Solution: solve for x

  5. Example #2 • Solve the equation. • x2 + 12x + 36 = 28

  6. Student Guided Practice • Solve the following equations • 1. 4x2 – 20 = 5 • 2. x2 + 8x + 16 = 49

  7. Completing the square • What is completing the square? • Answer: • If a quadratic expression of the form x2 + bxcannot model a square, you can add a term to form a perfect square trinomial. This is called completing the square.

  8. Completing the square

  9. Completing the square steps

  10. Example #3 • Solve the equation by completing the square. • x2= 12x – 20

  11. Example#4 • Solve the equation by completing the square • 18x + 3x2 = 45

  12. Example #5 • Solve each equation by completing the square. • 1) problem #1 in completing the square worksheet • p2 + 14p − 38 = 0

  13. Student guided practice • Do problems from worksheet 2-6 and 13-16

  14. Writing quadratic functions in vertex form • Recall the vertex form of a quadratic function from lesson: f(x) = a(x – h)2 + k, where the vertex is (h, k). • You can complete the square to rewrite any quadratic function in vertex form.

  15. Example #6 • Write the function in vertex form, and identify its vertex. • f(x) = x2 + 16x – 12

  16. Example#7 • Write the function in vertex form, and identify its vertex • g(x) = 3x2 – 18x + 7

  17. Student guided practice • Write the function in vertex form, and identify its vertex • g(x) = 3x2 – 18x + 7

  18. homework • Do problems 26 through 36 • From page 89

  19. Closure • Today we learn about solving a quadratic equation using completing the square. • Next class we are going to learn about complex numbers and roots

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