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Chapter 11

Chapter 11. Quadratic Functions and Equations. The Quadratic Formula. 11.2. Solving Using the Quadratic Formula Approximating Solutions. Solving Using the Quadratic Formula.

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Chapter 11

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  1. Chapter 11 Quadratic Functions and Equations

  2. The Quadratic Formula 11.2 • Solving Using the Quadratic Formula • Approximating Solutions

  3. Solving Using the Quadratic Formula Each time we solve by completing the square, the procedure is the same. When a procedure is repeated many times, a formula can often be developed to speed up our work. If we begin with a quadratic equation in standard form, ax2 + bx + c = 0, and solve by completing the square we arrive at the quadratic formula.

  4. The Quadratic Formula The solutions of ax2 + bx + c = 0, are given by

  5. Solve 3x2 + 5x = 2 using the quadratic formula. Solution First determine a, b, and c: 3x2 + 5x – 2 = 0; a = 3, b = 5, and c = –2. Substituting

  6. The solutions are 1/3 and –2. The check is left to the student.

  7. To Solve a Quadratic Equation 1. If the equation can easily be written in the form ax2 = p or (x + k)2 = d, use the principle of square roots. 2. If step (1) does not apply, write the equation in the form ax2 + bx + c = 0. 3. Try factoring using the principle of zero products. 4. If factoring seems difficult or impossible, use the quadratic formula. Completing the square can also be used. The solutions of a quadratic equation can always be found using the quadratic formula. They cannot always be found by factoring.

  8. Recall that a second-degree polynomial in one variable is said to be quadratic. Similarly, a second-degree polynomial function in one variable is said to be a quadratic function.

  9. Solve x2 + 7 = 2x using the quadratic formula. Solution First determine a, b, and c: x2 – 2x + 7 = 0; a = 1, b = –2, and c = 7. Substituting

  10. The solutions are The check is left to the student.

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