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MTH 065 Elementary Algebra II

MTH 065 Elementary Algebra II. Chapter 11 Quadratic Functions and Equations Section 11.1 Quadratic Equations. Geometric Representation of Completing the Square. x. x + 8. Area = x(x + 8). Geometric Representation of Completing the Square. x. x 2. 8x. x. 8. Area = x 2 + 8x.

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MTH 065 Elementary Algebra II

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  1. MTH 065Elementary Algebra II Chapter 11 Quadratic Functions and Equations Section 11.1 Quadratic Equations

  2. Geometric Representation ofCompleting the Square x x + 8 Area = x(x+ 8)

  3. Geometric Representation ofCompleting the Square x x2 8x x 8 Area = x2 + 8x

  4. Geometric Representation ofCompleting the Square x x2 8x x 8 Area = x2 + 8x

  5. Geometric Representation ofCompleting the Square x x2 8x x 4 4 Area = x2 + 8x

  6. Geometric Representation ofCompleting the Square x x2 4x 4x x 4 4 Area = x2 + 8x

  7. Geometric Representation ofCompleting the Square x x2 4x 4x x 4 4 Area = x2 + 8x

  8. Geometric Representation ofCompleting the Square 4 4x x x2 4x x 4 Area = x2 + 8x

  9. Geometric Representation ofCompleting the Square 4 4x ? x x2 4x x 4 Area = x2 + 8x + ?

  10. Geometric Representation ofCompleting the Square 4 4x 16 x x2 4x x 4 Area = x2 + 8x + 16

  11. Geometric Representation ofCompleting the Square 4 4x 16 x x2 4x x 4 Area = x2 + 8x + 16 = (x + 4)2

  12. Terminology • Quadratic Equation Any equation equivalent to an equation with the form … ax2 + bx + c = 0 … where a, b, & c are constants and a ≠ 0. • Quadratic Function Any function equivalent to the form … f(x) = ax2 + bx + c ... where a, b, & c are constants and a ≠ 0.

  13. Review Results from Chapter 6 • Solve quadratic equations by graphing. • Put into standard form: ax2 + bx + c = 0 • Graph the function: f(x) = ax2 + bx + c • Solutions are the x-intercepts. • # of Solutions? 0, 1, or 2 Details of Graphs of Quadratic Functions – Section 11.6

  14. Review Results from Chapter 6 • Solve quadratic equations by factoring. • Put into standard form: ax2 + bx + c = 0 • Factor the quadratic: (rx + m)(sx + n) = 0 • Set each factor equal to zero and solve. • # of Solutions? • 0  does not factor (not factorable  no solution) • 1  factors as a perfect square (if it factors) • 2  two different factors (if it factors)

  15. Principle of Square Roots For any non-negative real number k, if … … then …

  16. Principle of Square Roots For any non-negative real number k, if … … then … • Why? Consider the following example … • x2 = 9 •  x2 – 9 = 0 •  (x – 3)(x + 3) = 0 •  x = 3, –3

  17. Application of thePrinciple of Square Roots Solve the equation … Note This example demonstrates how to solve a quadratic equation with no linear (bx) term.

  18. Application of thePrinciple of Square Roots Solve the equation … Since the left side is positive and the right side is negative; there is no solution.

  19. Application of thePrinciple of Square Roots Solve the equations …

  20. Application of thePrinciple of Square Roots Solve the equation … But this does not factor …

  21. Solving by“Completing the Square” Note: This polynomial does not factor.

  22. Solving ax2 + bx + c = 0 by“Completing the Square” • Basic Steps … • Get into the form: ax2 + bx = d • Divide through by a giving: x2 + mx = n • Add the square of half of m to both sides. • i.e. add • Factor the left side (a perfect square). • Solve using the Principle of Square Roots.

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