Solve Quadratic Equations by Using the Quadratic Formula. Return to Table of Contents
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1 Solve Quadratic Equations by Using the Quadratic Formula Return to Table of Contents 128
2 Solving Quadratics At this point you have learned how to solve quadratic equations by: graphing factoring using square roots and completing the square Many quadratic equations may be solved using these methods. Though completing the square works for any quadratic equation, it can be cumbersome to repeatedly use the algorithm. Today we will be given a tool to solve ANY quadratic equation, and it ALWAYS works! 129
3 Completing the Square Now try Completing the Square on the standard form of a quadratic equation. Step 1 - Rewrite Equation Step 2 - Find (b/2) 2 Step 3 - Add the result to both sides Step 4 - Factor & Simplify Teacher Notes Step 5 - Take Square Root of both sides Step 6 - Write 2 Equations & Solve 130
4 Completing the Square Step 1 - Rewrite Equation and factor out a Steps 2 and 3 - Find (b/2) 2, Add the result to both sides, simplify Step 4 - Factor & Simplify Step 5 - Take Square Root of both sides Step 6 - Solve for x 131
5 The Quadratic Formula The solutions of ax 2 + bx + c = 0, where a 0, are: Teacher Notes 132
6 Quadratic Formula Example 1: Solve 2x 2 + 3x - 5 = 0 Once you identify the values of a, b, and c, simply substitute into the quadratic formula and simplify as much as possible. a = 2 b = 3 c = -5 Answer How can you check your answers? 133
7 Quadratic Formula Example 2: Solve: 2x = x 2-3 To use the Quadratic Formula, the equation must be in standard form (ax 2 + bx +c = 0). Identify a, b, and c, then substitue into the formula and simplify. Answer Don't forget to check your results! 134
8 44 Solve the following equation using the quadratic formula: Answer A -5 B -4 C -3 D -2 E -1 F 1 G 2 H 3 I 4 J 5 135
9 45 Solve the following equation using the quadratic formula: A -5 B -4 C -3 D -2 E -1 F 1 G 2 H 3 I 4 J 5 Answer 136
10 46 Solve the following equation using the quadratic formula: A -5 B -4 C -3 D -2 E -1.5 F 1.5 G 2 H 3 I 4 J 5 Answer 137
11 Quadratic Formula Example 3: Solve using the quadratic formula, and simplify the result. x 2-2x - 4 = 0 Answer 138
12 47 Find the larger solution to the equation. Answer 139
13 48 Find the smaller solution to the equation. Answer 140
14 Which Method Work in small groups to solve the quadratic equation using the following different methods. Factoring Quadratic Formula Completing the Square Graphing Teacher Notes 141
15 Which Method Work in small groups to solve the quadratic equation using the following different methods. Factoring Quadratic Formula Completing the Square Graphing Teacher Notes 142
16 The Discriminant Return to Table of Contents 143
17 Solutions Recall what it means to have 0, 1, or 2 solutions/zeros/roots 2 real solutions no real solutions 1 real solution 144
18 The Discriminant At times, it is not necessary to solve for the zeros or roots of a quadratic function, but simply to know how many roots exist (zero, one, or two). The quickest way to determine how many solutions a quadratic has, algebraically, is to calculate what's called the discriminant. It may look familiar, as the discriminant is a part of the quadratic formula. 145
19 The Discriminant Discriminant: the radicand in the Quadratic Formula (the piece of the equation under the radical sign). Note, it does NOT include the radical sign. Quadratic Formula: Discriminant: 146
20 The Discriminant Other important tips before practice: The square root of a positive number has two solutions. The square root of zero is 0. The square root of a negative number has no real solution. 147
21 The Discriminant Examine the following graphs and determine the relationship between the discriminant of a quadratic and its graph. Teacher Notes Discriminant Discriminant Discriminant 148
22 The Discriminant CONCLUSION: If b 2-4ac > 0 (POSITIVE) the quadratic has two real solutions click to reveal If b 2-4ac = 0 (ZERO) the quadratic has one real solution If b 2-4ac < 0 (NEGATIVE) the quadratic has no real solutions 149
23 49 What is the value of the discriminant of 2x 2-2x + 3 = 0? Answer 150
24 50 Use the discriminant to find the number of solutions for 2x 2-2x + 3 = 0 A 0 B 1 Answer C 2 151
25 51 What is the value of the discriminant of x 2-8x + 4 = 0? Answer 152
26 52 Use the discriminant to find the number of solutions for x 2-8x + 4 = 0 A 0 B 1 Answer C 2 153
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