Copyright © Cengage Learning. All rights reserved. Quadratic Equations, Quadratic Functions, and Complex Numbers 9

Copyright © Cengage Learning. All rights reserved. Section 9.3 Solving Quadratic Equations by the Quadratic Formula

3 Objectives 1.Use the quadratic formula to find all real solutions of a given quadratic equation. 2.Determine whether a quadratic equation has real solutions. 3.Solve an application involving a quadratic equation

4 Solving Quadratic Equations by the Quadratic Formula We can solve any quadratic equation by the method of completing the square, but the work is often tedious. In this section, we will develop a formula, called the quadratic formula, that will enable us to solve quadratic equations with much less effort.

5 Use the quadratic formula to find all real solutions of a given quadratic equation 1.

6 Use the quadratic formula to find all real solutions of a given quadratic equation We can solve the general quadratic equation by completing the square. ax 2 + bx + c = 0 Divide both sides by a. Simplify. Subtract from both sides. Standard form of a quadratic equation.

7 Use the quadratic formula to find all real solutions of a given quadratic equation We can now complete the square on x by adding, or, to both sides: After factoring the trinomial on the left side and adding the fractions on the right side, we have

8 Use the quadratic formula to find all real solutions of a given quadratic equation This equation can be solved by the square-root method to obtain

9 Use the quadratic formula to find all real solutions of a given quadratic equation These solutions usually are written in one expression called the quadratic formula. Quadratic Formula The solutions of the quadratic equation ax 2 + bx + c = 0 are

10 Example Solve using the quadratic formula: x 2 + 5x + 6 = 0. Solution: In this equation, a = 1, b = 5, and c = 6. We substitute these values into the quadratic formula and simplify. Substitute 1 for a, 5 for b, and 6 for c. Simplify.

11 Example – Solution Thus, The solutions are –2 and –3. Check both solutions. Because the solutions were rational, this equation could have been solved by factoring. or Subtract. cont’d

12 Determine whether a quadratic equation has real-number solutions 2.

13 Example Determine whether the equation x 2 + 2x + 5 = 0 has real-number solutions. Solution: In this equation, a = 1, b = 2, and c = 5. We substitute these values into the quadratic formula. Substitute 1 for a, 2 for b, and 5 for c.

14 Example – Solution Since is not a real number, there are no real-number solutions. cont’d

15 Solve an application involving a quadratic equation 3.

16 Example – Manufacturing Samsung Electronics manufactures a 46-inch wide-screen television set. The 46-inch screen is measured along the diagonal. The screens are rectangular in shape and are 17 inches wider than they are high. Find the dimensions of a screen. Analyze the problem We can let h represent the height of a screen in inches, as shown in Figure 9-2. Then h + 17 will represent the width in inches. Figure 9-2

17 Example – Manufacturing Form an equation Since the sides of the screen and its diagonal form a right triangle, we can use the Pythagorean theorem to form the equation h 2 + (h + 17) 2 = 46 2 which we can solve as follows. Solve the equation h 2 + h h = 2,116 2h h – 1,827 = 0 Subtract 2,116 from both sides. Square each term.

18 Example – Manufacturing We can solve this equation with the quadratic formula. cont’d

19 Example – Manufacturing cont’d or

20 Example – Manufacturing State the conclusion The height of each screen will be approximately 22.9 inches, and the width will be approximately or 39.9 inches. We discard the second solution, because the height of a TV screen cannot be negative. Check the result The width of 39.9 inches is 17 inches wider than the height of 22.9 inches. (22.9) 2 + (39.9) 2  46 2 cont’d