Section 2.3: Quadratic Equations
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1 SECTION.3 Quadratic Equations Section.3: Quadratic Equations Solving by Factoring Solving by Completing the Square Solving by the Quadratic Formula The Discriminant Solving by Factoring MATH 1310 College Algebra 97
2 CHAPTER Solving Equations and Inequalities Additional Example 1: 98 University of Houston Department of Mathematics
3 SECTION.3 Quadratic Equations Additional Example : Additional Example 3: MATH 1310 College Algebra 99
4 CHAPTER Solving Equations and Inequalities Additional Example 4: 100 University of Houston Department of Mathematics
5 SECTION.3 Quadratic Equations Additional Example 5: Area = (5 ft)(8 ft) = 40 sq ft MATH 1310 College Algebra 101
6 CHAPTER Solving Equations and Inequalities Solving by Completing the Square 10 University of Houston Department of Mathematics
7 SECTION.3 Quadratic Equations Additional Example 1: MATH 1310 College Algebra 103
8 CHAPTER Solving Equations and Inequalities Additional Example : Additional Example 3: 104 University of Houston Department of Mathematics
9 SECTION.3 Quadratic Equations Additional Example 4: MATH 1310 College Algebra 105
10 CHAPTER Solving Equations and Inequalities Solving by the Quadratic Formula 106 University of Houston Department of Mathematics
11 SECTION.3 Quadratic Equations Additional Example 1: MATH 1310 College Algebra 107
12 CHAPTER Solving Equations and Inequalities Additional Example : 108 University of Houston Department of Mathematics
13 SECTION.3 Quadratic Equations Additional Example 3: Additional Example 4: MATH 1310 College Algebra 109
14 CHAPTER Solving Equations and Inequalities = University of Houston Department of Mathematics
15 SECTION.3 Quadratic Equations The Discriminant MATH 1310 College Algebra 111
16 CHAPTER Solving Equations and Inequalities 11 University of Houston Department of Mathematics
17 SECTION.3 Quadratic Equations Additional Example 1: Additional Example : MATH 1310 College Algebra 113
18 CHAPTER Solving Equations and Inequalities Additional Example 3: Additional Example 4: 114 University of Houston Department of Mathematics
19 Exercise Set.3: Quadratic Equations Solve the following equations by factoring. 1. x 10x + 1 = 0. x + 13x + 40 = 0 3. x ( x ) = x ( x + 8) = 0 5. x + 14x = 7 6. x 60 = 11x 7. x 7x 15 = x 7x + 4 = x + 17x = x 7x = x 5 = 0 1. x 49 = x 9 = x 5 = 0 Solve the following equations by factoring. To simplify the process, remember to first factor out the Greatest Common Factor (GCF). 15. x 8x = x + 10x = x + 1x = x 30x = x 1 = x + 7 = x 15x + 90 = 0. 4x + 0x + 4 = x + 30x 30 = x 75x + 18 = 0 Find all real solutions of the following equations by completing the square. 5. x + 8x + 1 = 0 6. x 6x 40 = 0 7. x ( x 10) = x + 4x = 8 9. x + 14x + 60 = x ( x + 1) = x + 5x 5 = 0 3. x 7x + = x 1x 5 = x + 1x = x + 8x = x 40x 78 = x 5x = 38. 8x 6x 5 = 0 Find all real solutions of the following equations by using the quadratic formula. 39. x + 5x + = x 7x + 3 = x 6x + 8 = 0 4. x x = x + 5x + 7 = x 8x = x + 10x + 5 = x 6x + 5 = x 5x = x + 3x 6 = x 6x 3 = x + 8x 1 = 0 MATH 1310 College Algebra 115
20 Exercise Set.3: Quadratic Equations Find all real solutions of the following equations by using a method of your choice. 51. x 10x + 16 = 0 5. x 6x 8 = x 4x = x + 1x = 36 Use the discriminant to determine the number of real solutions of each equation. (Do not solve the equation.) 55. x 5x + = x + 3x + 7 = x + 4x = x 6x = x + 4x = x + 7x = x cx d = 0, where d > x rx + t = 0, where t < 0 Use the discriminant to find all values of k so that each of the following equations has exactly one solution. 63. kx + 6x + 3k = x + kx + 49 = The height of a triangle is 3 cm shorter than its base. If the area of the triangle is 90 cm, find the base and height of the triangle. 68. Find x if the area of the figure below is 6cm. (Note that the figure may not be drawn to scale.) x cm 3 cm x cm 69. Find two consecutive odd integers that have a product of Find two numbers that have a sum of 39 and a product of John can paint the fence in 3 hours less time than Chris. If it takes them 5 4 of an hour when working together, how long does it take each of them to paint the fence individually? 7. Marie takes 1 hour more time to clean the house than Tina. If it takes them hours when working together, how long would it take each of them to clean the house individually? For each of the following problems: (a) Model the situation by writing appropriate equation(s). (b) Solve the equation(s) and then answer the question posed in the problem. 65. The length of a rectangular frame is 5 cm longer than its width. If the area of the frame is 36 cm, find the length and width of the frame. 66. The height of a right triangle is 4 inches longer than its base. If its diagonal measures 0 inches, find the base and the height of the triangle. 116 University of Houston Department of Mathematics
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