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Copyright © 2012 Pearson Education, Inc 8.2 The Quadratic Formula ■ Solving Using the Quadratic Formula ■ Approximating Solutions

Slide 8- 3 Copyright © 2012 Pearson Education, Inc Solving Using the Quadratic Formula Each time we solve by completing the square, the procedure is the same. In mathematics, 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, ax 2 + bx + c = 0, and solve by completing the square we arrive at the quadratic formula.

Slide 8- 4 Copyright © 2012 Pearson Education, Inc The Quadratic Formula The solutions of ax 2 + bx + c = 0, are given by

Slide 8- 5 Copyright © 2012 Pearson Education, Inc Example Solution Solve 3x 2 + 5x = 2 using the quadratic formula. First determine a, b, and c: 3x 2 + 5x – 2 = 0; a = 3, b = 5, and c = –2. Substituting

Slide 8- 6 Copyright © 2012 Pearson Education, Inc The solutions are 1/3 and –2. The check is left to the student.

Slide 8- 7 Copyright © 2012 Pearson Education, Inc To Solve a Quadratic Equation 1.If the equation can easily be written in the form ax 2 = 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 ax 2 + 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, but is usually slower.

Slide 8- 8 Copyright © 2012 Pearson Education, Inc 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.

Slide 8- 9 Copyright © 2012 Pearson Education, Inc Example Solution First determine a, b, and c: x 2 – 2x + 7 = 0; a = 1, b = –2, and c = 7. Solve x = 2x using the quadratic formula. Substituting

Slide Copyright © 2012 Pearson Education, Inc The solutions are The check is left to the student.

Slide Copyright © 2012 Pearson Education, Inc Approximating Solutions When the solution of an equation is irrational, a rational-number approximation is often useful. This is often the case in real- world applications similar to those found in section 8.3.

Slide Copyright © 2012 Pearson Education, Inc Example Solution Use a calculator to approximate Take the time to familiarize yourself with your calculator: