150. a. Clear fractions in the following equation and write in. b. For the equation you wrote in part (a), compute. The Quadratic Formula

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1 75 CHAPTER Quadratic Equations and Functions Preview Eercises Eercises 8 50 will help you prepare for the material covered in the net section. 8. a. Solve by factoring: b. The quadratic equation in part (a) is in the standard form a + b + c 0. Compute b - ac. Is b - ac a perfect square? 9. a. Solve by factoring: b. The quadratic equation in part (a) is in the standard form a + b + c 0. Compute b - ac. 50. a. Clear fractions in the following equation and write in the form a + b + c 0: + -. b. For the equation you wrote in part (a), compute b - ac. SECTION. The Quadratic Formula Objectives Solve quadratic equations using the quadratic formula. Use the discriminant to determine the number and type of solutions. Determine the most efficient method to use when solving a quadratic equation. 5 Write quadratic equations from solutions. Use the quadratic formula to solve problems. Until fairly recently, many doctors believed that your blood pressure was theirs to know and yours to worry about. Today, however, people are encouraged to find out their blood pressure. That pumped-up cuff that squeezes against your upper arm measures blood pressure in millimeters (mm) of mercury (Hg). Blood pressure is given in two numbers: systolic pressure over diastolic pressure, such as 0 over 80. Systolic pressure is the pressure of blood against the artery walls when the heart contracts. Diastolic pressure is the pressure of blood against the artery walls when the heart is at rest. In this section, we will derive a formula that will enable you to solve quadratic equations more quickly than the method of completing the square. Using this formula, we will work with functions that model changing systolic pressure for men and women with age. Solve quadratic equations using the quadratic formula. Solving Quadratic Equations Using the Quadratic Formula We can use the method of completing the square to derive a formula that can be used to solve all quadratic equations. The derivation given on the net page also shows a particular quadratic equation, - - 0, to specifically illustrate each of the steps. ISBN Introductory & Intermediate Algebra for College Students, Third Edition, by Robert Blitzer. Published by Prentice Hall. Copyright 009 by Pearson Education, Inc.

2 SECTION. The Quadratic Formula 75 Deriving the Quadratic Formula Standard Form of a Quadratic Equation Comment A Specific Eample a + b + c 0, a b a + c a 0 This is the given equation. Divide both sides by the coefficient of b a - c a Isolate the binomial by adding - c on both sides. a - + b a + a b a b q half - c a + a b a b Complete the square. Add the square of half the coefficient of to both sides. - + a - b q half + a - b + b a + a + b a b b a - c a + b a - c a # a a + b a Factor on the left side and obtain a common denominator on the right side a - b # + 9 a + b a b a + b a b -ac + b a b - ac a Add fractions on the right side. a - b a - b b a ; b - ac B a + b a ; b - ac a -b a ; b - ac a Apply the square root property. Take the square root of the quotient, simplifying the denominator. b Solve for by subtracting from both sides. a - ; A 9 - ; ; -b ; b - ac a Combine fractions on the right side. ; The formula shown at the bottom of the left column is called the quadratic formula. A similar proof shows that the same formula can be used to solve quadratic equations if a, the coefficient of the -term, is negative. The Quadratic Formula The solutions of a quadratic equation in standard form a Z 0, are given by the quadratic formula: a + b + c 0, with b _ b -ac. a equals negative b plus or minus the square root of b ac, all divided by a. ISBN To use the quadratic formula, write the quadratic equation in standard form if necessary. Then determine the numerical values for a (the coefficient of the -term), b (the coefficient of the -term), and c (the constant term). Substitute the values of a, b, and c into the quadratic formula and evaluate the epression. The ; sign indicates that there are two (not necessarily distinct) solutions of the equation. Introductory & Intermediate Algebra for College Students, Third Edition, by Robert Blitzer. Published by Prentice Hall. Copyright 009 by Pearson Education, Inc.

3 75 CHAPTER Quadratic Equations and Functions EXAMPLE Solve using the quadratic formula: Solving a Quadratic Equation Using the Quadratic Formula Solution a, b, and c. The given equation is in standard form. Begin by identifying the values for a 8 b c Using Technology Graphic Connections The graph of the quadratic function y has -intercepts at - and This verifies that E -, F is the solution set of the quadratic equation y 8 +. Substituting these values into the quadratic formula and simplifying gives the equation s solutions. -b ; b - ac a - ; ; ; ; 6 6 Use the quadratic formula. Substitute the values for a, b, and c: a 8, b, and c -. Now we will evaluate this epression in two different ways to obtain the two solutions. On the left, we will add 6 to -. On the right, we will subtract 6 from or The solutions are and and the solution set is E -, F. -, -intercept is. -intercept is. [,, ] by [, 0, ] In Eample, the solutions of are rational numbers. This means that the equation can also be solved by factoring. The reason that the solutions are rational numbers is that b - ac, the radicand in the quadratic formula, is 6, which is a perfect square. If a, b, and c are rational numbers, all quadratic equations for which b - ac is a perfect square have rational solutions. CHECK POINT Solve using the quadratic formula: EXAMPLE Solve using the quadratic formula: Solving a Quadratic Equation Using the Quadratic Formula +. Solution The quadratic equation must be in standard form to identify the values for a, b, and c. To move all terms to one side and obtain zero on the right, we subtract + from both sides. Then we can identify the values for a, b, and c. ISBN Introductory & Intermediate Algebra for College Students, Third Edition, by Robert Blitzer. Published by Prentice Hall. Copyright 009 by Pearson Education, Inc.

4 SECTION. The Quadratic Formula This is the given equation. Subtract + from both sides. a b c Substituting these values into the quadratic formula and simplifying gives the equation s solutions. Using Technology You can use a graphing utility to verify that the solutions of are ; 6. Begin by entering y - - in the Y screen. Then evaluate this function at each of the proposed solutions. -b ; b - ac a -- ; ; ; 6 ; ; 6 ; 6 Use the quadratic formula. Substitute the values for a, b, and c: a, b -, and c # 6 # 6 6 Factor out from the numerator. Divide the numerator and denominator by. In each case, the function value is 0, verifying that the solutions satisfy 0. ; 6 The solutions are, and the solution set is b + 6, - 6 r or b ; 6 r. In Eample, the solutions of + are irrational numbers. This means that the equation cannot be solved by factoring. The reason that the solutions are irrational numbers is that b - ac, the radicand in the quadratic formula, is, which is not a perfect square. Notice, too, that the solutions, and, are conjugates. Study Tip Many students use the quadratic formula correctly until the last step, where they make an error in simplifying the solutions. Be sure to factor the numerator before dividing the numerator and the denominator by the greatest common factor. ISBN ; 6 You cannot divide just one term in the numerator and the denominator by their greatest common factor. _ 6 _ 6 ; 6 ; 6 Incorrect! _ 6 _ 6 ; 6 Introductory & Intermediate Algebra for College Students, Third Edition, by Robert Blitzer. Published by Prentice Hall. Copyright 009 by Pearson Education, Inc.

5 756 CHAPTER Quadratic Equations and Functions Can all irrational solutions of quadratic equations be simplified? No. The following solutions cannot be simplified: 5_ 7 Other than, terms in each numerator have no common factor.. _ 7 CHECK POINT Solve using the quadratic formula: 6 -. EXAMPLE Solve using the quadratic formula: Solution Solving a Quadratic Equation Using the Quadratic Formula + -. Begin by writing the quadratic equation in standard form This is the given equation. Add to both sides. a b c Using Technology Graphic Connections The graph of the quadratic function y + + has no -intercepts. This verifies that the equation in Eample, + -, or + + 0, has imaginary solutions. Substituting these values into the quadratic formula and simplifying gives the equation s solutions. -b ; b - ac a - ; - # # # - ; ; ; i 6 - ; i 6 - ; i - ; i Use the quadratic formula. Substitute the values for a, b, and c: a, b, and c. Multiply under the radical. Subtract under the radical i8 i # i Factor out from the numerator. Divide the numerator and denominator by. Epress in the form a + bi, writing i before the square root. y + + [,, ] by [, 0, ] The solutions are - and the solution set is b - or + i, - - i ; i, r b - ; i r. In Eample, the solutions of + - are imaginary numbers. This means that the equation cannot be solved using factoring. The reason that the solutions are imaginary numbers is that b - ac, the radicand in the quadratic formula, is -8, which is negative. Notice, too, that the solutions are comple conjugates. ISBN Introductory & Intermediate Algebra for College Students, Third Edition, by Robert Blitzer. Published by Prentice Hall. Copyright 009 by Pearson Education, Inc.

6 SECTION. The Quadratic Formula 757 CHECK POINT Solve using the quadratic formula: Some rational equations can be solved using the quadratic formula. For eample, consider the equation + -. The denominators are and. The least common denominator is. We clear fractions by multiplying both sides of the equation by. Notice that cannot equal zero. b a+ a b, 0 + # Use the distributive property. Simplify. By adding to both sides of + -, quadratic equation: we obtain the standard form of the Use the discriminant to determine the number and type of solutions. This is the equation that we solved in Eample. The two imaginary solutions are not part of the restriction that Z 0. The Discriminant The quantity b - ac, which appears under the radical sign in the quadratic formula, is called the discriminant. Table. shows how the discriminant of the quadratic equation a + b + c 0 determines the number and type of solutions. Study Tip Checking irrational and imaginary solutions can be time-consuming. The solutions given by the quadratic formula are always correct, unless you have made a careless error. Checking for computational errors or errors in simplification is sufficient. Table. The Discriminant and the Kinds of Solutions to a b c 0 Discriminant Kinds of Solutions Graph of b ac to a b c 0 y a b c b - ac 7 0 Two unequal real solutions: If a, b, and c are rational numbers and the discriminant is a perfect square, the solutions are rational. If the discriminant is not a perfect square, the solutions are irrational conjugates. y b - ac 0 One solution (a repeated solution) that is a real numbers: If a, b, and c are rational numbers, the repeated solution is also a rational number. Two -intercepts y One -intercept ISBN b - ac 6 0 No real solution; two imaginary solutions: The solutions are comple conjugates. y No -intercepts Introductory & Intermediate Algebra for College Students, Third Edition, by Robert Blitzer. Published by Prentice Hall. Copyright 009 by Pearson Education, Inc.

7 758 CHAPTER Quadratic Equations and Functions EXAMPLE Using the Discriminant For each equation, compute the discriminant. Then determine the number and type of solutions: a. b. c Study Tip The discriminant is b - ac. It is not b - ac, so do not give the discriminant as a radical. Solution Begin by identifying the values for a, b, and c in each equation. Then compute b - ac, the discriminant. a a b c 5 Substitute and compute the discriminant: b - ac - # The discriminant, 76, is a positive number that is not a perfect square. Thus, there are two real irrational solutions. (These solutions are conjugates of each other.) b a 9 b 6 c Substitute and compute the discriminant: The discriminant, 0, shows that there is only one real solution. This real solution is a rational number. c a b 8 c 7 b - ac -6 - # 9 # b - ac -8 - # # The negative discriminant, -0, shows that there are two imaginary solutions. (These solutions are comple conjugates of each other.) CHECK POINT For each equation, determine the number and type of solutions: a. b. c compute the discriminant. Then Determine the most efficient method to use when solving a quadratic equation. Determining Which Method to Use All quadratic equations can be solved by the quadratic formula. However, if an equation is in the form u d, such as 5 or + 8, it is faster to use the square root property, taking the square root of both sides. If the equation is not in the form u d, write the quadratic equation in standard form a + b + c 0. Try to solve the equation by factoring. If a + b + c cannot be factored, then solve the quadratic equation by the quadratic formula. Because we used the method of completing the square to derive the quadratic formula, we no longer need it for solving quadratic equations. However, we will use completing the square in Chapter to help graph certain kinds of equations. ISBN Introductory & Intermediate Algebra for College Students, Third Edition, by Robert Blitzer. Published by Prentice Hall. Copyright 009 by Pearson Education, Inc.

8 SECTION. The Quadratic Formula 759 Table. summarizes our observations about which technique to use when solving a quadratic equation. Table. Determining the Most Efficient Technique to Use When Solving a Quadratic Equation Description and Form of Most Efficient the Quadratic Equation Solution Method Eample a + b + c 0 and a + b + c can be factored easily. a + c 0 The quadratic equation has no -term. b 0 u d; u is a first-degree polynomial. a + b + c 0 and a + b + c cannot be factored or the factoring is too difficult. Factor and use the zero-product principle. Solve for and apply the square root property. Use the square root property. Use the quadratic formula: -b ; b - ac. a or ; a b c ;5 - ; 5 -- ; ; - -6 ; 8 ; 7 ; 7 ; 7 ; 7 Write quadratic equations from solutions. Writing Quadratic Equations from Solutions Using the zero-product principle, the equation has two solutions, and -5. By applying the zero-product principle in reverse, we can find a quadratic equation that has two given numbers as its solutions. The Zero-Product Principle in Reverse If A 0 or B 0, then AB 0. ISBN EXAMPLE 5 Writing Equations from Solutions Write a quadratic equation with the given solution set: a. e - 5 b. 5-5i, 5i6., f Introductory & Intermediate Algebra for College Students, Third Edition, by Robert Blitzer. Published by Prentice Hall. Copyright 009 by Pearson Education, Inc.

9 760 CHAPTER Quadratic Equations and Functions Solution a. Because the solution set is e - 5 then, f, - 5 or. NOT AVAILABLE FOR ELECTRONIC VIEWING The special properties of zero make it possible to write a quadratic equation from its solutions. + 5 Obtain zero on one side of each 0 or - 0 equation or - 0 Clear fractions, multiplying by and, respectively Use the zero-product principle in reverse: If A 0 or B 0, then AB Use the FOIL method to multiply. Combine like terms. Thus, one equation is Many other quadratic equations have E - 5, F for their solution sets. These equations can be obtained by multiplying both sides of by any nonzero real number. b. Because the solution set is 5-5i, 5i6, then -5i or 5i. + 5i 0 or - 5i 0 + 5i - 5i 0-5i 0-5i Obtain zero on one side of each equation. Use the zero-product principle in reverse: If A 0 or B 0, then AB 0. Multiply conjugates using A + BA - B A - B. 5i 5 i 5i i - This is the required equation. CHECK POINT 5 Write a quadratic equation with the given solution set: E - 5, F a. b. 5-7i, 7i6. 5 Use the quadratic formula to solve problems. Applications Quadratic equations can be solved to answer questions about variables contained in quadratic functions. EXAMPLE 6 Blood Pressure and Age The graphs in Figure.6 illustrate that a person s normal systolic blood pressure, measured in millimeters of mercury (mm Hg), depends on his or her age.the function PA 0.006A - 0.0A + 0 models a man s normal systolic pressure, PA, at age A. a. Find the age, to the nearest year, of a man whose normal systolic blood pressure is 5 mm Hg. b. Use the graphs in Figure.6 to describe the differences between the normal systolic blood pressures of men and women as they age. Normal Blood Pressure (mm Hg) P(A) Normal Systolic Blood Pressure and Age Men Women Age FIGURE.6 A ISBN Introductory & Intermediate Algebra for College Students, Third Edition, by Robert Blitzer. Published by Prentice Hall. Copyright 009 by Pearson Education, Inc.

10 SECTION. The Quadratic Formula 76 Solution a. We are interested in the age of a man with a normal systolic blood pressure of 5 millimeters of mercury. Thus, we substitute 5 for PA in the given function for men. Then we solve for A, the man s age. P(A)0.006A -0.0A A -0.0A A -0.0A-5 a b 0.0 c 5 This is the given function for men. Substitute 5 for PA. Subtract 5 from both sides and write the quadratic equation in standard form. Because the trinomial on the right side of the equation is prime, we solve using the quadratic formula. Notice that the variable is A, rather than the usual. A b_ b -ac a ( 0.0)_ ( 0.0) -(0.006)( 5) (0.006) Use the quadratic formula. Substitute the values for a, b, and a 0.006, b -0.0, and c -5. c: Using Technology On most calculators, here is how to approimate 0.0 ; Use a calculator to simplify the epression under the square root Many Scientific Calculators L 0.0 ; Use a calculator: 0.0 L ,.0 Many Graphing Calculators ( ) A L A L or A L A 7 Reject this solution. Age cannot be negative Use a calculator and round to the nearest integer. ISBN ENTER If your calculator displays an open parenthesis after, you ll need to enter another closed parenthesis here. The positive solution, A L, indicates that is the approimate age of a man whose normal systolic blood pressure is 5 mm Hg. This is illustrated by the black lines with the arrows on the red graph representing men in Figure.7. b. Take a second look at the graphs in Figure.7. Before approimately age 50, the blue graph representing women s normal systolic blood pressure lies below the red graph representing men s normal systolic blood pressure. Thus, up to age 50, women s normal systolic blood pressure is lower than men s, Normal Blood Pressure (mm Hg) although it is increasing at a faster rate. After age 50, women s normal systolic blood pressure is higher than men s P(A) FIGURE.7 Normal Systolic Blood Pressure and Age Blood pressure: 5 Men Women Age ~ Age A Introductory & Intermediate Algebra for College Students, Third Edition, by Robert Blitzer. Published by Prentice Hall. Copyright 009 by Pearson Education, Inc.

11 76 CHAPTER Quadratic Equations and Functions CHECK POINT 6 The function PA 0.0A A + 07 models a woman s normal systolic blood pressure, PA, at age A. Use this function to find the age, to the nearest year, of a woman whose normal systolic blood pressure is 5 mm Hg. Use the blue graph in Figure.7 on the previous page to verify your solution.. EXERCISE SET Practice Eercises In Eercises 6, solve each equation using the quadratic formula. Simplify solutions, if possible In Eercises 9 0, compute the discriminant. Then determine the number and type of solutions for the given equation In Eercises 6, solve each equation by the method of your choice. Simplify solutions, if possible In Eercises 7 60, write a quadratic equation in standard form with the given solution set , , e -, f ISBN Introductory & Intermediate Algebra for College Students, Third Edition, by Robert Blitzer. Published by Prentice Hall. Copyright 009 by Pearson Education, Inc.

12 SECTION. The Quadratic Formula e - 5 6, f i, 6i i, 8i , , , , i, - i i, - i , , When the sum of 6 and twice a positive number is subtracted from the square of the number, 0 results. Find the number. 66. When the sum of and twice a negative number is subtracted from twice the square of the number, 0 results. Find the number. In Eercises 67 7, solve each equation by the method of your choice Practice PLUS Eercises 6 6 describe quadratic equations. Match each description with the graph of the corresponding quadratic function. Each graph is shown in a [-0, 0, ] by [-0, 0, ] viewing rectangle. 6. A quadratic equation whose solution set contains imaginary numbers 6. A quadratic equation whose discriminant is 0 6. A quadratic equation whose solution set is E ; F 6. A quadratic equation whose solution set contains integers a. b. c. Fatal Crashes per 00 Million Miles Driven ƒ ƒ ƒ + ƒ Application Eercises A driver s age has something to do with his or her chance of getting into a fatal car crash. The bar graph shows the number of fatal vehicle crashes per 00 million miles driven for drivers of various age groups. For eample, 5-year-old drivers are involved in. fatal crashes per 00 million miles driven. Thus, when a group of 5-year-old Americans have driven a total of 00 million miles, approimately have been in accidents in which someone died. Age of United States Drivers and Fatal Crashes Source: Insurance Institute for Highway Safety Age of Drivers ISBN d. The number of fatal vehicle crashes per 00 million miles, f, for drivers of age can be modeled by the quadratic function f Use the function to solve Eercises What age groups are epected to be involved in fatal crashes per 00 million miles driven? How well does the function model the trend in the actual data shown in the bar graph? 7. What age groups are epected to be involved in 0 fatal crashes per 00 million miles driven? How well does the function model the trend in the actual data shown in the bar graph? Introductory & Intermediate Algebra for College Students, Third Edition, by Robert Blitzer. Published by Prentice Hall. Copyright 009 by Pearson Education, Inc.

13 76 CHAPTER Quadratic Equations and Functions Throwing events in track and field include the shot put, the discus throw, the hammer throw, and the javelin throw. The distance that an athlete can achieve depends on the initial velocity of the object thrown and the angle above the horizontal at which the object leaves the hand. Angle at which the shot is released Path of shot Distance Achieved Path's maimum horizontal distance In Eercises 75 76, an athlete whose event is the shot put releases the shot with the same initial velocity, but at different angles. 75. When the shot is released at an angle of 5, its path can be modeled by the function f , 78. The length of a rectangle eceeds twice its width by inches. If the area is 0 square inches, find the rectangle s dimensions. Round to the nearest tenth of an inch. 79. The longer leg of a right triangle eceeds the shorter leg by inch, and the hypotenuse eceeds the longer leg by 7 inches. Find the lengths of the legs. Round to the nearest tenth of a inch. 80. The hypotenuse of a right triangle is 6 feet long. One leg is feet shorter than the other. Find the lengths of the legs. Round to the nearest tenth of a foot. 8. A rain gutter is made from sheets of aluminum that are 0 inches wide. As shown in the figure, the edges are turned up to form right angles. Determine the depth of the gutter that will allow a cross-sectional area of square inches. Show that there are two different solutions to the problem. Round to the nearest tenth of an inch. Flat sheet 0 inches wide 0 in which is the shot s horizontal distance, in feet, and f is its height, in feet. This function is shown by one of the graphs, (a) or (b), in the figure. Use the function to determine the shot s maimum distance. Use a calculator and round to the nearest tenth of a foot.which graph, (a) or (b), shows the shot s path? 8. A piece of wire is 8 inches long. The wire is cut into two pieces and then each piece is bent into a square. Find the length of each piece if the sum of the areas of these squares is to be square inches. 8 (a) 8 Height (b) 8 inches 8 Horizontal Distance [0, 80, 0] by [0, 0, 0] 76. When the shot is released at an angle of 65, its path can be modeled by the function f , in which is the shot s horizontal distance, in feet, and f is its height, in feet. This function is shown by one of the graphs, (a) or (b), in the figure above. Use the function to determine the shot s maimum distance. Use a calculator and round to the nearest tenth of a foot. Which graph, (a) or (b), shows the shot s path? 77. The length of a rectangle is meters longer than the width. If the area is 8 square meters, find the rectangle s dimensions. Round to the nearest tenth of a meter. cut 8. Working together, two people can mow a large lawn in hours. One person can do the job alone hour faster than the other person. How long does it take each person working alone to mow the lawn? Round to the nearest tenth of an hour. 8. A pool has an inlet pipe to fill it and an outlet pipe to empty it. It takes hours longer to empty the pool than it does to fill it. The inlet pipe is turned on to fill the pool, but the outlet pipe is accidentally left open. Despite this, the pool fills in 8 hours. How long does it take the outlet pipe to empty the pool? Round to the nearest tenth of an hour. Writing in Mathematics 85. What is the quadratic formula and why is it useful? 86. Without going into specific details for every step, describe how the quadratic formula is derived. ISBN Introductory & Intermediate Algebra for College Students, Third Edition, by Robert Blitzer. Published by Prentice Hall. Copyright 009 by Pearson Education, Inc.

14 SECTION. The Quadratic Formula Eplain how to solve using the quadratic formula. 88. If a quadratic equation has imaginary solutions, how is this shown on the graph of the corresponding quadratic function? 89. What is the discriminant and what information does it provide about a quadratic equation? 90. If you are given a quadratic equation, how do you determine which method to use to solve it? 9. Eplain how to write a quadratic equation from its solution set. Give an eample with your eplanation. Technology Eercises 9. Use a graphing utility to graph the quadratic function related to any five of the quadratic equations in Eercises 9 0. How does each graph illustrate what you determined algebraically using the discriminant? 9. Reread Eercise 8. The cross-sectional area of the gutter is given by the quadratic function f 0 -. Graph the function in a [0, 0, ] by [0, 60, 5] viewing rectangle. Then TRACE along the curve or use the maimum function feature to determine the depth of the gutter that will maimize its cross-sectional area and allow the greatest amount of water to flow. What is the maimum area? Does the situation described in Eercise 8 take full advantage of the sheets of aluminum? 00. In using the quadratic formula to solve the quadratic equation 5-7, we have a 5, b, and c The quadratic formula can be used to solve the equation Solve for t: s -6t + v 0 t. 0. A rectangular swimming pool is meters long and 8 meters wide.a tile border of uniform width is to be built around the pool using 0 square meters of tile. The tile is from a discontinued stock (so no additional materials are available) and all 0 square meters are to be used. How wide should the border be? Round to the nearest tenth of a meter. If zoning laws require at least a -meter-wide border around the pool, can this be done with the available tile? 0. The area of the shaded region outside the rectangle and inside the triangle is 0 square yards. Find the triangle s height, represented by. Round to the nearest tenth of a yard. + Critical Thinking Eercises Makes Sense? In Eercises 9 97, determine whether each statement makes sense or does not make sense and eplain your reasoning. Review Eercises Solve: ƒ 5 + ƒ ƒ - ƒ. (Section 9., Eample ) ISBN Because I want to solve fairly quickly, I ll use the quadratic formula I simplified to + because is a factor of. 96. I need to find a square root to determine the discriminant. 97. I obtained -7 for the discriminant, so there are two imaginary irrational solutions. In Eercises 98 0, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. 98. Any quadratic equation that can be solved by completing the square can be solved by the quadratic formula. 99. The quadratic formula is developed by applying factoring and the zero-product principle to the quadratic equation a + b + c Solve: (Section 0.6, Eample ) 07. Rationalize the denominator: 5 +. (Section 0.5, Eample 5) Preview Eercises Eercises 08 0 will help you prepare for the material covered in the net section. 08. Use point plotting to graph f and g + in the same rectangular coordinate system. 09. Use point plotting to graph f and g + in the same rectangular coordinate system. 0. Find the -intercepts for the graph of f Introductory & Intermediate Algebra for College Students, Third Edition, by Robert Blitzer. Published by Prentice Hall. Copyright 009 by Pearson Education, Inc.