1. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Section 15.3 Quadratic Equations: The Quadratic Formula

2. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. General Form of a Quadratic Equation The general quadratic equation is ax 2 + bx + c = 0 where a, b, and c are real constants and a ≠ 0. Quadratic Formula The solutions of the general quadratic equation ax 2 + bx + c = 0, where a ≠ 0, are

3. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Notes The expression b 2  4ac is called the discriminant. If The Quadratic Formula

4. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. The Quadratic Formula Solve the following quadratic equations by using the quadratic formula: 2x 2 + x – 2 = 0

5. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. The Quadratic Formula In many practical applications of quadratic equations, we want to know a decimal approximation of the solutions. Using a calculator, we find the following approximate values to the solutions of the equation 2x 2 + x – 2 = 0:

6. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. The Quadratic Formula

7. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. The Quadratic Formula This shows that the quadratic formula works correctly even though the leading coefficient a is negative. We could also multiply all the terms on both sides of the equation by  1 and solve the new equation. The solutions will be the same.

8. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. The Quadratic Formula

9. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. The Quadratic Formula

10. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. The Quadratic Formula Use the quadratic formula to solve the following quadratic equation.

11. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. The Quadratic Formula Note: Whenever the solutions are rational numbers, the equation can be solved by factoring. In this example, we could have solved as follows.

12. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. The Quadratic Formula Solve the following equation:

13. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Non-real Solutions Solve the following equation: x 2 + x + 1 = 0