CHAPTER 5 The Quadratic Formula

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Clementine Higgins
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1 CHAPTER 5 The Quadratic Formula The main focus of t his chapt er is t o learn how t o use one of t he most basic, most widely used and most important formulas in algebra namely the quadratic formula. We will develop this formula in section 5.2 and use it to solve a variet y of problems in this and in subsequent chapters. The technique developed in sect ion 5.1 t o solve quadrat ics is mainly used to develop the quadratic formula given in sect ions 5.2. Section 5.1 Solving Quadratic Equations by Completing the Square Key Concept s: completing the square Goals. Our goal in t his sect ion is t o develop t he process commonly ref erred to as completing the square and show how it can be used to solve quadratic equat ions. A positive integer is called a perfect square if it can be rewritten as a number times itself. 25 = 5 5, 16 =, and 6 = 8 8 are all perfect squares (numbers). A trinomial is called a perfect square (trinomial) if it can be expressed as a binomial t imes it self. x 2 + x + = (x + 2) (x + 2), x 2-6x + 9 = (x - 3) (x - 3), and x 2-1x + 9 = (x - 7) (x - 7) are all perfect square t rinomials. To complete the square (for a trinomial) means to add a constant to the given expression so that the new expression is a perfect square. Procedure for completing t he square for expressions of t he t ype x 2 + bx is to: First, comput e ( 1 2 b )2 Second, add ( 1 2 b )2 to x 2 + bx Example 1. Complet e t he square on x 2 + 6x

2 Take 1 2 of the coefficient of x and square it: x 2 + 6x x 2 + 6x of 6 is 3 and 32 = 9 add 9 t o t he given expression x 2 + 6x + 9 = (x + 3) 2 is a perfect square t rinomial. Example 2. Complet e t he square on y 2 + 7y Take 1 2 of the coefficient of y and square it: y 2 + 7y y 2 + 7y of 7 is 7 2 and ( 7 2 )2 = 9 add 9 to the given expression y 2 + 7y + 9 = (y )2 is a perfect square t rinomial. From Chapt er 3 we recall t hat a short met hod of solving t he quadrat ic equat ion y 2 = 7 is to take the square root of both sides remembering to prefix the by both a " + " sign and a " - " sign. The answers to the equation y 2 = 7 are y = + 7 and y = - 7 or more simply y = ± 7. Example 3. Solve t he equat ion (x + 2) 2 = 7 (x + 2) 2 = 7 take the square root of both sides, remembering t o prefix the symbol by both a + and a - sign x + 2 = ± 7 add -2 to both sides x = -2 ± 7 or x = and x = -2-7 are t he t wo solut ions or approximat ely x = =.6 and x = = -.6 The previous example gives us a hint on how t o solve a quadrat ic equat ion which cannot be solved by factoring as we did in chapter 3 The given equat ion in example 3 was of t he form:

3 perfect square = a constant (*) Our object ive will be t o t ake a quadrat ic equat ion in st andard f orm and t o complete the square of the variable terms so as to obtain an equation in form (*) Example. Solve t he equat ion x 2 + 6x - 8 = 0 x 2 + 6x - 8 = 0 x 2 + 6x = 8 x 2 + 6x + 9 = rewrite add 8 t o bot h sides complete the square on the left side by ( )2 = 9 Remember 9 must be added t o both sides to maintain equality (x + 3) 2 = 17 take the square root of both sides, remembering to prefix the symbol by both a + and a - sign x + 3 = ± 17 add -3 to both sides x = -3 ± 17 or x = and x = are t he t wo solut ions or approximat ely x = = 1.1 and x = = -7.1 Example 5. Solve t he equat ion x 2-5x - 7 = 0 x 2-5x - 7 = 0 x 2-5x = 7 x 2-5x + 25 = add 7 t o bot h sides ( )2 = 25 Remember 25 must be added t o bot h sides t o maint ain equalit y rewrite (x - 2.5) 2 = take the square root of both sides x = ± add 2.5 t o bot h sides x = 2.5 ± or x = and x = are t he t wo solut ions or approximat ely

4 x = = 6.1 and x = = -1.1 Example 6. Solve t he equat ion 2x 2 + x - 5 = 0 x 2 + 2x = 0 divide both sides by 2 x 2 + 2x = 0 add 5 2 to both sides x 2 + 2x = 5 2 ( )2 = 1. Remember: 1 must be added to both sides to maintain equality x 2 + 2x + 1 = rewrite (x + 1) 2 = 3.5 take the square root of both sides, remembering to prefix symbol by both a + and a - sign x + 1 = ± 3.5 add -1 to both sides x = -1 ± 3.5 or x = and x = are t he t wo solut ions or approximat ely x = =.87 or x = = Example 7. Solve t he equat ion 3x 2-5x + 3 = 0 3x 2-5x + 3 = 0 divide both sides by 3 x x + 1 = 0 x x = -1 add -1 to both sides x x + 25 = ( )2 = Remember must be added t o bot h sides t o maintain equality (x )2 = - 11 There are no real solut ions t o t his equat ion. In Sect ion 5.3, we will int roduce the complex numbers, which include numbers whose squares are negative

5 Procedure for solving a quadrat ic equat ion of t he form ax 2 + bx + c = 0, a 0 by the method of completing the square (1) Divide both sides of the equation by a if a 1 (2) The resulting equation will be of the form x 2 + dx + e = 0. Take 1 2 of the coefficient of x and square it to obtain ( d 2 )2 (3) Add ( d 2 )2 to both sides of the equation. () Take t he square root of bot h sides and solve for x in bot h of t he result ing equat ions

Solving Quadratic Equations

Solving Quadratic Equations MATH 101 College Algebra J. Robert Buchanan Department of Mathematics Summer 2012 Objectives In this lesson we will learn to: solve quadratic equations by factoring, solve quadratic