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Understanding the Quadratic Formula: Solving Quadratic Functions and Equations

This chapter explores quadratic functions and equations, focusing on the quadratic formula as a tool for finding solutions. By starting with a quadratic equation in standard form (ax² + bx + c = 0), we derive the quadratic formula through the method of completing the square. The chapter also outlines steps to solve quadratic equations, including factoring and using the quadratic formula when necessary. Several examples illustrate the application of the formula, enhancing understanding and mastery of solving quadratic equations.

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Understanding the Quadratic Formula: Solving Quadratic Functions and Equations

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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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