Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1 1 Chapter 9 Quadratic Equations and Functions
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 2 Quadratic Equations and Functions CHAPTER 9.1The Square Root Principle and Completing the Square 9.2Solving Quadratic Functions Using the Quadratic Formula 9.3Solving Equations That Are Quadratic in Form 9.4Graphing Quadratic Functions 9 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 2
3 The Square Root Principle and Completing the Square Use the square root principle to solve quadratic equations. 2.Solve quadratic equations by completing the square.
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 4 The Square Root Principle If x 2 = a, where a is a real number, then It is common to indicate the positive and negative solutions by writing Note: The expression is read “plus or minus the square root of a.”
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 5 Example Solve. Solution Check Use the square root principle. Simplify by factoring out a perfect square. Check True. Note: Remember the ± means that the two solutions are and. True.
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 6 Example Solve. Solution Use the square root principle.
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 7 Example Solve. Solution Subtract 14 from both sides to isolate x 2. Use the square root principle. Simplify by factoring out a perfect square.
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 8 To solve equations in the form ax 2 = b, we first isolate x 2 by dividing both sides of the equation by a. We solve an equation in the form ax 2 + b = c by using both the addition and multiplication principles of equality to isolate x 2 before using the square root principle. In an equation in the form (ax + b) 2 = c, notice the expression ax + b is squared. We can use the square root principle to eliminate the square.
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 9 Example Solve. Solution Subtract 5 from both sides. Use the square root principle. Simplify by factoring out a perfect square. Divide both sides by 7.
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 10 Example Solve. Solution Add 3 to both sides and divide each side by 5, to isolate x. Use the square root principle. or
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 11 Example Solve. Solution Add 1 to both sides. Use the square root principle. Divide both sides by 4 to solve for x.
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 12 Example Solve by completing the square. Solution We first write the equation in the form x 2 + bx = c. Complete the square by adding 25 to both sides. Add 16 to both sides to get the form x 2 + bx = c. Factor. Use the square root principle. Subtract 5 from both sides to isolate x.
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 13 Example Solution Complete the square by adding 36 to both sides. Subtract 9 from both sides to get the form x 2 + bx = c. Factor. Use the square root principle. Add 6 to both sides to isolate x. Simplify the square root. Solve by completing the square.
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 14 Solving Quadratic Equations by Completing the Square To solve a quadratic equation by completing the square: 1. Write the equation in the form x 2 + bx = c. 2. Complete the square by adding (b/2) 2 to both sides. 3. Write the completed square in factored form and simplify the right side, 4. Use the square root principle to eliminate the square. 5. Isolate the variable. 6.Simplify as needed.
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 15 Example Solve by completing the square. Solution Divide both sides by 2. Simplify. Add to both sides to complete the square.
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 16 continued Combine the fractions. Factor. Add to both sides and simplify the square root. Use the square root principle.
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 17 Example Solve by completing the square. Solution
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 18 continued