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

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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

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

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

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

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:

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

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.

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

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

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.

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.

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:

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