6.9 Solving Quadratic Equations by Using Factoring and by Using the Quadratic Formula 383 In the quadratic formula, the plus or minus symbol, ,± is used. If, for example, x 2 3, = ± then x 2 3 5 = + = or x 2 3 1. = − = − It is possible for a quadratic equation to have no real solution. When solving a quadratic equation, if the radicand (the expression inside the square root) is a negative number, the quadratic equation has no real solution . To use the quadratic formula, first write the quadratic equation in standard form. Then determine the values of a (the coefficient of the squared term), b (the coefficient of the x term), and c (the constant). Finally, substitute the values of a b, , and c into the quadratic formula and evaluate. Quadratic Formula For a quadratic equation in standard form, ax bx c a 0, 0, 2 + + = ≠ the quadratic formula is x b b ac a 4 2 2 = − ± − Did You Know? The Mathematics of Motion The free fall of an object is something that has interested scientists and mathematicians for centuries. It is described by a quadratic equation. Shown here is a time-lapse photo that shows the free fall of a ball in equal time intervals. What you see can be described verbally this way: The rate of change in velocity in each interval is the same; therefore, velocity is continuously increasing and acceleration is constant. Example 10 Using the Quadratic Formula Solve the equation x x2 15 0 2 − − = by using the quadratic formula. Solution In this equation, a b c 1, 2, and 15. = = − = − = − ± − = − − ± − − − = ± + = ± = ± x b b ac a 4 2 ( 2) ( 2) 4(1)( 15) 2(1) 2 4 60 2 2 64 2 2 8 2 2 2 + = = − = − = − 2 8 2 10 2 5 or 2 8 2 6 2 3 The solutions are 5 and 3. − 7 Now try Exercise 61 Example 11 Irrational Solutions to a Quadratic Equation Solve x x 4 8 1 2 − = − by using the quadratic formula. Solution Begin by writing the equation in standard form by adding 1 to both sides of the equation, which gives the following. x x 4 8 1 0 2 − + = = = − = a b c 4, 8, 1 = − ± − = − − ± − − x b b ac a 4 2 ( 8) ( 8) 4(4)(1) 2(4) 2 2 Note that Example 10 can also be solved by factoring. Sciencephotos/Alamy Stock Photo
RkJQdWJsaXNoZXIy NjM5ODQ=