9 Solving Quadratic Equations
99.19.29.39.49.59.6Solving QuadraticEquationsProperties of RadicalsSolving Quadratic Equations by GraphingSolving Quadratic Equations Using Square RootsSolving Quadratic Equations by Completing the SquareSolving Quadratic Equations Using the Quadratic FormulaSolving Nonlinear Systems of EquationsDolphinDl hi ((p. 521)Half-pipeHlf i ((p. 513)PondPd ((p. 501)SEE the Big IdeaKicker (p. 493)ParthenonPPartarthhenonhenon (p.(p 48483)3)hsnb alg1 pe 09op.indd 4762/5/15 8:55 AM
Maintaining Mathematical ProficiencyFactoring Perfect Square TrinomialsExample 1 Factor x2 14x 49.x2 14x 49 x2 2(x)(7) 72 (x 7)2Write as a2 2ab b2.Perfect square trinomial patternFactor the trinomial.1. x2 10x 252. x2 20x 1003. x2 12x 364. x2 18x 815. x2 16x 646. x2 30x 225Solving Systems of Linear Equations by GraphingExample 2 Solve the system of linear equations by graphing.y 2x 1Equation 11y — x 83Equation 2Step 1 Graph each equation.yStep 2 Estimate the point of intersection.The graphs appear to intersect at (3, 7).6Step 3 Check your point from Step 2.Equation 1Equation 2y 2x 11y —x 83? 17 —(3) 837 2(3) 17 7 7 713y x 84y 2x 12 2246 x The solution is (3, 7).Solve the system of linear equations by graphing.7. y 5x 3y 2x 438. y —2 x 21y —4 x 519. y —2 x 4y 3x 310. ABSTRACT REASONING What value of c makes x2 bx c a perfect square trinomial?Dynamic Solutions available at BigIdeasMath.comhsnb alg1 pe 09op.indd 4774772/5/15 8:55 AM
MathematicalPracticesMathematically proficient students monitor their work and changecourse as needed.Problem-Solving StrategiesCore ConceptGuess, Check, and ReviseWhen solving a problem in mathematics, it is often helpful to estimate a solutionand then observe how close that solution is to being correct. For instance, you canuse the guess, check, and revise strategy to find a decimal approximation of thesquare root of 2.1.2.3.Guess1.41.411.415Check1.42 1.961.412 1.98811.4152 2.002225How to reviseIncrease guess.Increase guess.Decrease guess.By continuing this process, you can determine that the square root of 2 isapproximately 1.4142.Approximating a Solution of an EquationThe graph of y x2 x 1 is shown.Approximate the positive solution of the equationx2 x 1 0 to the nearest thousandth.2 2SOLUTIONUsing the graph, you can make an initial estimate ofthe positive solution to be x 0.65.1.2.3.4.yGuess0.650.62Check0.652 0.65 1 0.07250.622 0.62 1 0.00440.6180.61810.6182 0.618 1 0.0000760.61812 0.6181 1 0.000152 x 2y x2 x 1How to reviseDecrease guess.Decrease guess.Increase guess.The solution is between 0.618 and 0.6181.So, to the nearest thousandth, the positive solution of the equation is x 0.618.Monitoring Progress1. Use the graph in Example 1 to approximate the negative solutionof the equation x2 x 1 0 to the nearest thousandth.1 32. The graph of y x2 x 3 is shown. Approximate both solutions 1of the equation x2 x 3 0 to the nearest thousandth.Chapter 9hsnb alg1 pe 09op.indd 478x 2 y 478yx2 x 3Solving Quadratic Equations2/5/15 8:55 AM
9.1Properties of RadicalsEssential QuestionHow can you multiply and divide square roots?Operations with Square RootsWork with a partner. For each operation with square roots, compare the resultsobtained using the two indicated orders of operations. What can you conclude?a. Square Roots and Addition———Is 36 64 equal to 36 64 ?———In general, is a b equal to a b ? Explain your reasoning.b. Square Roots and Multiplication— ——Is 4 9 equal to 4 9 ?— ——In general, is a b equal to a b ? Explain your reasoning.c. Square Roots and Subtraction———Is 64 36 equal to 64 36 ?———In general, is a b equal to a b ? Explain your reasoning.d. Square Roots and Division—REASONINGABSTRACTLYTo be proficient in math,you need to recognize anduse counterexamples. — 100100Is ——?— equal to4 4—— aaIn general, is —— ? Explain your reasoning.— equal tob b Writing CounterexamplesWork with a partner. A counterexample is an example that proves that a generalstatement is not true. For each general statement in Exploration 1 that is not true, writea counterexample different from the example given.Communicate Your Answer3. How can you multiply and divide square roots?4. Give an example of multiplying square roots and an example of dividing squareroots that are different from the examples in Exploration 1.5. Write an algebraic rule for each operation.a. the product of square rootsb. the quotient of square rootsSection 9.1hsnb alg1 pe 0901.indd 479Properties of Radicals4792/5/15 8:56 AM
9.1LessonWhat You Will LearnUse properties of radicals to simplify expressions.Simplify expressions by rationalizing the denominator.Core VocabulVocabularylarrycounterexample, p. 479radical expression, p. 480simplest form of a radical,p. 480rationalizing the denominator,p. 482conjugates, p. 482like radicals, p. 484Perform operations with radicals.Using Properties of RadicalsA radical expression is an expression that contains a radical. An expression involvinga radical with index n is in simplest form when these three conditions are met. No radicands have perfect nth powers as factors other than 1. No radicands contain fractions. No radicals appear in the denominator of a fraction.Previousradicandperfect cubeYou can use the property below to simplify radical expressions involving square roots.Core ConceptProduct Property of Square RootsWordsSTUDY TIPThere can be more thanone way to factor aradicand. An efficientmethod is to find thegreatest perfectsquare factor.The square root of a product equals the product of the square roots ofthe factors. —— ———Algebra ab a b , where a, b 0—Using the Product Property of Square Roots 36 3——a. 108 36 3—Factor using the greatest perfect square factor.—Product Property of Square Roots— 6 3—Simplify. 9 x —b. 9x3 9 x2 xSTUDY TIP—In this course, whenevera variable appears inthe radicand, assumethat it has onlynonnegative values.— 9 5 9 5 3 5Numbers—2Factor using the greatest perfect square factor.—Product Property of Square Roots x— 3x xSimplify.Monitoring ProgressHelp in English and Spanish at BigIdeasMath.comSimplify the expression.——2. 801. 24——4. 75n53. 49x3Core ConceptQuotient Property of Square RootsWordsThe square root of a quotient equals the quotient of the square roots ofthe numerator and denominator. —Numbers480Chapter 9hsnb alg1 pe 0901.indd 480——3 3 3 —— —4 —24 ab ab , where a 0 and b 0—Algebra—— ——Solving Quadratic Equations2/5/15 8:56 AM
Using the Quotient Property of Square Roots ——a. 151564— ——Quotient Property of Square Roots 64— 15 —8b.Simplify. 81x 81x———2——Quotient Property of Square Roots29 —xSimplify.You can extend the Product and Quotient Properties of Square Roots to other radicals,such as cube roots. When using these properties of cube roots, the radicands maycontain negative numbers.Using Properties of Cube Roots3— 64 23—a. 128 64 23—STUDY TIPTo write a cube root insimplest form, find factorsof the radicand that areperfect cubes.Factor using the greatest perfect cube factor.3—Product Property of Cube Roots3— 4 23—Simplify. 125 x x3—b. 125x7 125 x6 x3—3— —c.3 36Product Property of Cube Roots3—5x2 xSimplify.3— yy— —3—216 2163— y —6—d.Factor using the greatest perfect cube factors.3—8x427yQuotient Property of Cube RootsSimplify.3— 8x4—3 ——Quotient Property of Cube Roots3 27y3 8 x x —— 27 y3— 8 x3 x —3— 27 y33—3—3—33—Factor using the greatest perfect cube factors.Product Property of Cube Roots3—3—3x2x —3ySimplify.Monitoring ProgressHelp in English and Spanish at BigIdeasMath.comSimplify the expression. —5.23—93—9. 54 176. —1007. 10.3— 16x411. 8.3a— 274x264———Section 9.1hsnb alg1 pe 0901.indd 48136—z2 ———12.325c7d 364—Properties of Radicals4812/5/15 8:56 AM
Rationalizing the DenominatorWhen a radical is in the denominator of a fraction, you can multiply the fraction by anappropriate form of 1 to eliminate the radical from the denominator. This process iscalled rationalizing the denominator.Rationalizing the Denominator———— 5 5 3na. ——— ——— 3n 3n 3nSTUDY TIP 3nMultiply by ——. 3n—Rationalizing thedenominator worksbecause you multiplythe numerator anddenominator by the samenonzero number a, whichis the same as multiplyingaby —, or 1.a 15n —— 9n2Product Property of Square Roots 15n ——— 9 n2Product Property of Square Roots 15n —3nSimplify.— — 223—3— 3 3Multiply by —.3— 3 —b. ——3—3—3— 9 9 33—2 3 —3— 27Product Property of Cube Roots2 3 —3Simplify.3—————The binomials a b c d and a b c d , where a, b, c, and d are rational numbers,are called conjugates. You can use conjugates to simplify radical expressions thatcontain a sum or difference involving square roots in the denominator.Rationalizing the Denominator Using ConjugatesLOOKING FORSTRUCTURENotice that the product of——two conjugates a b c d——and a b c d does notcontain a radical and is arational number.————( a b c d )( a b c d )— 2— 2 ( a b ) ( c d )7Simplify ——.2 3SOLUTION7—2 3 2 3 2 37—— ——2 3————7( 2 3 ) —— 222 ( 3 )—14 7 3 —1— 14 7 3 a2b c2d—The conjugate of 2 3 is 2 3 .Sum and difference patternSimplify.Simplify.Monitoring ProgressHelp in English and Spanish at BigIdeasMath.comSimplify the expression.113. —— 5517. ——3 32482Chapter 9hsnb alg1 pe 0901.indd 482— 1014. —— 3818. ——1 3 —715. —— 2x— 1319. —— 5 216.2y23—1220. ——— 2 7Solving Quadratic Equations2/5/15 8:57 AM
Solving a Real-Life ProblemThe distance d (in miles) that you can see to the horizon with your eye level h feet—3habove the water is given by d — . How far can you see when your eye level is25 feet above the water? SOLUTION —3(5)d —2— 15 —— 25 ft—Substitute 5 for h.Quotient Property of Square Roots—— 15 2 ———— 2 2 2Multiply by ——. 2— 30 —2Simplify.— 30You can see —, or about 2.74 miles.2Modeling with Mathematics—31 mhThe ratio of the length to the width of a golden rectangle is ( 1 5 ) : 2. Thedimensions of the face of the Parthenon in Greece form a golden rectangle. What isthe height h of the Parthenon?SOLUTION1. Understand the Problem Think of the length and height of the Parthenon asthe length and width of a golden rectangle. The length of the rectangular face is31 meters. You know the ratio of the length to the height. Find the height h.—2. Make a Plan Use the ratio ( 1 5 ) : 2 to write a proportion and solve for h.—3. Solve the Problem1 5 312h—h( 1 5 ) 62— —Write a proportion.Cross Products Property62h ——1 5—621 5h ————1 5 1 5—62 62 5h — 4h 19.16 —Divide each side by 1 5 .Multiply the numerator anddenominator by the conjugate.Simplify.Use a calculator.The height is about 19 meters.—1 5314. Look Back — 1.62 and — 1.62. So, your answer is reasonable.19.162Monitoring ProgressHelp in English and Spanish at BigIdeasMath.com21. WHAT IF? In Example 6, how far can you see when your eye level is 35 feet abovethe water?22. The dimensions of a dance floor form a golden rectangle. The shorter side of thedance floor is 50 feet. What is the length of the longer side of the dance floor?Section 9.1hsnb alg1 pe 0901.indd 483Properties of Radicals4832/5/15 8:57 AM
Performing Operations with RadicalsSTUDY TIPDo not assume thatradicals with differentradicands cannot beadded or subtracted.Always check to seewhether you can simplifythe radicals. In some cases,the radicals will becomelike radicals.Radicals with the same index and radicand are called like radicals. You can add andsubtract like radicals the same way you combine like terms by using the DistributiveProperty.Adding and Subtracting Radicals——————a. 5 7 11 8 7 5 7 8 7 11Commutative Property of Addition—— (5 8) 7 11Distributive Property—— 3 7 11——Subtract. 10 5 4 5————b. 10 5 20 10 5 4 5—Factor using the greatest perfect square factor.—Product Property of Square Roots— 10 5 2 5Simplify.— (10 2) 5Distributive Property— 12 5——Add.—333c. 6 x 2 x (6 2) xDistributive Property—3 8 xAdd.Multiplying Radicals———Simplify 5 ( 3 75 ).SOLUTION————Method 1 5 ( 3 75 ) 5 3 5 75——————— 15 375Product Property of Square Roots 15 5 15Simplify.— (1 5) 15Distributive Property— 4 15————Subtract.———Method 2 5 ( 3 75 ) 5 ( 3 5 3 )—Simplify 75 .— 5 [ (1 5) 3 ]—Distributive Property— 5 ( 4 3 )Subtract.— 4 15Monitoring ProgressDistributive PropertyProduct Property of Square RootsHelp in English and Spanish at BigIdeasMath.comSimplify the expression.———23. 3 2 6 10 23—3—25. 4 5x 11 5x—227. ( 2 5 4 )484Chapter 9hsnb alg1 pe 0901.indd 484——24. 4 7 6 63———26. 3 ( 8 2 7 32 )3—3—3—28. 4 ( 2 16 )Solving Quadratic Equations2/5/15 8:57 AM
9.1ExercisesDynamic Solutions available at BigIdeasMath.comVocabulary and Core Concept Check1. COMPLETE THE SENTENCE The process of eliminating a radical from the denominator of aradical expression is called .—2. VOCABULARY What is the conjugate of the binomial 6 4?13— 2x9—3. WRITING Are the expressions — 2x and — equivalent? Explain your reasoning.4. WHICH ONE DOESN’T BELONG? Which expression does not belong with the other three?Explain your reasoning.—— —13 66 31 ———6 3 3 3Monitoring Progress and Modeling with MathematicsIn Exercises 5–12, determine whether the expressionis in simplest form. If the expression is not in simplestform, explain why.——15. 196.—7 ——7. 488. 34—3 1010. —459. —— 2111. ——32 23—12. 6 54In Exercises 13–20, simplify the expression.(See Example 1.)—13. 20—15. 128—17. 125b—19. 81m3 49423 64 49a 1004x—3—hsnb alg1 pe 0901.indd 485—2 81y 1000x33.6c— 125—20. 48n5 8173 38. 36.321 64a b—3 6—— 72 4 18—— 4 18— 2 18 ——33—128y3 128y3 —12512564 2 y3 3———1253— 3—3— 64 2 y3 ——125——3—4y 2 —125—26.8h427—ERROR ANALYSIS In Exercises 37 and 38, describe andcorrect the error in simplifying the expression.—65 121 144k 25v363—2—322. —24.34.—35. ——3——2——27.3—32. 343n218. 4x2—25.3———23.3—30. 10831. 64x516. 72——3—29. 1637.—14. 32In Exercises 21–28, simplify the expression.(See Example 2.)21.In Exercises 29–36, simplify the expression.(See Example 3.)28.2—Section 9.1Properties of Radicals4852/5/15 8:57 AM
In Exercises 39– 44, write a factor that you can use torationalize the denominator of the expression.4139. ——40. —— 13z 63m241. ——42. ——43. ——44. ——— 3 7362. MODELING WITH MATHEMATICS The orbital periodof a planet is the time it takes the planet to travelaround the Sun. You can find the orbital period P—(in Earth years) using the formula P d 3 , whered is the average distance (in astronomical units,abbreviated AU) of the planet from the Sun.3 4 x2— 2 5 85JupiterSunIn Exercises 45–54, simplify the expression.(See Example 4.)2445. —— 246. —— 3—47. ——48. 484—52b. What is Jupiter’s orbital period?63. MODELING WITH MATHEMATICS The electric1349. ——50. —— a a. Simplify the formula.— 5 2x—51.d 5.2 AU— 83d 2552. ——— 3n3 —453. ——54.3 2531108y—2current I (in amperes) an appliance uses is given by—Pthe formula I — , where P is the power (in watts)Rand R is the resistance (in ohms). Find the current anappliance uses when the power is 147 watts and theresistance is 5 ohms. In Exercises 55– 60, simplify the expression.(See Example 5.)155. —— 7 1256. ——5 3—— 1057. ——7 2 558. ——6 5—359. ——— 5 2 360. ——— 7 361. MODELING WITH MATHEMATICS The time t (inseconds) it takes an object to hit the ground is given—hby t — , where h is the height (in feet) from which16the object was dropped. (See Example 6.) a. How long does it take an earring to hit the groundwhen it falls from the roof of the building?b. How much soonerdoes the earring hitthe ground when it isdropped from two stories(22 feet) below the roof?486Chapter 9hsnb alg1 pe 0901.indd 48655 ft64. MODELING WITH MATHEMATICS You can find theaverage annual interest rate r (in decimal form) of—Va savings account using the formula r —2 1,V0where V0 is the initial investment and V2 is thebalance of the account after 2 years. Use the formulato compare the savings accounts. In which accountwould you invest money? Explain. AccountInitialinvestmentBalance after2 years1 275 2932 361 3823 199 2144 254 2725 386 406Solving Quadratic Equations2/5/15 8:57 AM
In Exercises 65– 68, evaluate the function for the givenvalue of x. Write your answer in simplest form and indecimal form rounded to the nearest hundredth.——65. h(x) 5x ; x 1066. g(x) 3x ; x 60 3x 3x 6x 1;x 8p(x) 5x—;x 467. r(x) —2————————86. 7y ( 27y 5 12y )2( 4 —2 —98 )88.——( —3 —48 )( 20 5 )3——3——33— 390. 2 ( 135 4 5 )91. MODELING WITH MATHEMATICS The circumferenceC of the art room in a mansion is approximated by—a2 b2the formula C 2π — . Approximate the2circumference of the room. —70. 4c 6ab——84. 3 ( 72 3 2 )85. 5 ( 2 6x 96x )3—In Exercises 69–72, evaluate the expression whena 2, b 8, and c —12 . Write your answer in simplestform and in decimal form rounded to the nearesthundredth.——89. 3 ( 4 32 )—69. a2 bc——83. 2 ( 45 5 )87.—68.In Exercises 83–90, simplify the expression.(See Example 9.)—71. 2a2 b272. b2 4ac73. MODELING WITH MATHEMATICS The text in thea 20 ftbook shown forms a golden rectangle. What is thewidth w of the text? (See Example 7.)b 16 ftentenntrrancentraentrancehalllivingroomhall6 in.diningroomguestroomguestroomw in.74. MODELING WITH MATHEMATICS The flag of Togo isapproximately the shape of a golden rectangle. Whatis the width w of the flag?92. CRITICAL THINKING Determine whether eachexpression represents a rational or an irrationalnumber. Justify your answer.42 in.—a. 4 6 48b. —— 38c. —— 12d. 3 7—w in.——ae. ——— , where a is a positive integer 10 2—2 5f. —— , where b is a positive integer2b 5b2In Exercises 75–82, simplify the expression.(See Example 8.)———75. 3 2 2 6 2——77. 2 6 5 54—79. 12 6 3 2 63—3—81. 81 4 3——————78. 9 32 2———76. 5 5 13 8 5—80. 3 7 5 14 2 283—3—82. 6 128t 2 2tIn Exercises 93–98, simplify the expression.93. 5x1395. 256y—5—96. 160x65—4—97. 6 9 9 3 9Section 9.1hsnb alg1 pe 0901.indd 48745—4—4— 1081—94.—55—4—5—98. 2 ( 7 16 )Properties of Radicals4872/5/15 8:57 AM
REASONING In Exercises 99 and 100, use the table102. HOW DO YOU SEE IT? The edge length s of a cube isshown.1—420— 3— 3an irrational number, the surface area is an irrationalnumber, and the volume is a rational number. Give apossible value of s.π21—4s0s— 3s— 3π103. REASONING Leta and b be positive numbers.—Explain why ab lies between a and b on a numberline. (Hint: Let a b and multiply each side of a bby a. Then let a b and multiply each side by b.)99. Copy and complete the table by (a) finding each()sum 2 2, 2 —14 , etc. and (b) finding each product( 2 2, 2 1—4 ,)104. MAKING AN ARGUMENT Your friend saysetc. .that you can rationalize the denominator of the2expression —by multiplying the numerator3—4 53—and denominator by 4 5 . Is your friend correct?Explain.100. Use your answers in Exercise 99 to determine whethereach statement is always, sometimes, or never true.Justify your answer.a. The sum of a rational number and a rationalnumber is rational.105. PROBLEM SOLVING The ratio of consecutiveanterms —in the Fibonacci sequence gets closer andan 1—1 5closer to the golden ratio — as n increases. Find2the term that precedes 610 in the sequence.b. The sum of a rational number and an irrationalnumber is irrational.c. The sum of an irrational number and an irrationalnumber is irrational.d. The product of a rational number and a rationalnumber is rational.—1 52—1 5and the golden ratio conjugate — for each of the2following.106. THOUGHT PROVOKING Use the golden ratio —e. The product of a nonzero rational number and anirrational number is irrational.f. The product of an irrational number and anirrational number is irrational.a. Show that the golden ratio and golden ratioconjugate are both solutions of x2 x 1 0.101. REASONING Let m be a positive integer. For whatb. Construct a geometric diagram that has the goldenratio as the length of a part of the diagram.values of m will the simplified form of the expression— 2m contain a radical? For what values will it notcontain a radical? Explain.107. CRITICAL THINKING Use the special product pattern(a b)(a2 ab b2) a3 b3 to simplify the2expression —. Explain your reasoning.3— x 1Maintaining Mathematical ProficiencyReviewing what you learned in previous grades and lessonsGraph the linear equation. Identify the x-intercept. (Section 3.5)108. y x 4109. y 2x 61110. y —3 x 13111. y —2 x 6Solve the equation. Check your solution. (Section 6.5)112. 32 2x488Chapter 9hsnb alg1 pe 0901.indd 488113. 27x 3x 6114.(—)1 2x6 2161 x1 x 2( )115. 625x —25Solving Quadratic Equations2/5/15 8:57 AM
9.2Solving Quadratic Equationsby GraphingEssential QuestionHow can you use a graph to solve a quadraticequation in one variable?Based on what you learned about thex-intercepts of a graph in Section 3.4,it follows that the x-intercept of thegraph of the linear equationy ax b2 variablesy6The x-interceptof the graph ofy x 2 is 2.4ax b 0.1 variableThe solution of theequation x 2 0is x 2.2is the same value as the solution of 6 4You can use similar reasoning tosolve quadratic equations.2( 2, 0)46 x 2 4Solving a Quadratic Equation by GraphingWork with a partner.10a. Sketch the graph of y x2 2x.y8b. What is the definition of anx-intercept of a graph? How manyx-intercepts does this graph have?What are they?642c. What is the definition of a solutionof an equation in x? How manysolutions does the equationx2 2x 0 have? What are they? 6 4 2246x 2d. Explain how you can verify thesolutions you found in part (c). 4Solving Quadratic Equations by GraphingMAKING SENSEOF PROBLEMSTo be proficient in math,you need to check youranswers to problems usinga different method andcontinually ask yourself,“Does this make sense?”Work with a partner. Solve each equation by graphing.a. x2 4 0b. x2 3x 0c. x2 2x 0d. x2 2x 1 0e. x2 3x 5 0f. x2 3x 6 0Communicate Your Answer3. How can you use a graph to solve a quadratic equation in one variable?4. After you find a solution graphically, how can you check your resultalgebraically? Check your solutions for parts (a) (d) in Exploration 2algebraically.5. How can you determine graphically that a quadratic equation has no solution?Section 9.2hsnb alg1 pe 0902.indd 489Solving Quadratic Equations by Graphing4892/5/15 8:57 AM
9.2 LessonWhat You Will LearnSolve quadratic equations by graphing.Use graphs to find and approximate the zeros of functions.Core VocabulVocabularylarryquadratic equation, p. 490Previousx-interceptrootzero of a functionSolve real-life problems using graphs of quadratic functions.Solving Quadratic Equations by GraphingA quadratic equation is a nonlinear equation that can be written in the standard formax2 bx c 0, where a 0.In Chapter 7, you solved quadratic equations by factoring. You can also solve quadraticequations by graphing.Core ConceptSolving Quadratic Equations by GraphingStep 1 Write the equation in standard form, ax2 bx c 0.Step 2 Graph the related function y ax2 bx c.Step 3 Find the x-intercepts, if any.The solutions, or roots, of ax2 bx c 0 are the x-intercepts of the graph.Solving a Quadratic Equation: Two Real SolutionsSolve x2 2x 3 by graphing.SOLUTIONStep 1 Write the equation in standard form.x2 2x 3Write original equation.x2 2x 3 0Subtract 3 from each side.Step 2 Graph the related functiony x2 2x 3.2Step 3 Find the x-intercepts.The x-intercepts are 3 and 1. 4 2y2 x 2So, the solutions are x 3and x 1.y x 2 2x 3Checkx2 2x 3Original equation?( 3)2 2( 3) 33 3 Monitoring ProgressSubstitute.Simplify.x2 2x 3?12 2(1) 33 3 Help in English and Spanish at BigIdeasMath.comSolve the equation by graphing. Check your solutions.1. x2 x 2 0490Chapter 9hsnb alg1 pe 0902.indd 4902. x2 7x 103. x2 x 12Solving Quadratic Equations2/5/15 8:57 AM
Solving a Quadratic Equation: One Real SolutionSolve x2 8x 16 by graphing.SOLUTIONStep 1 Write the equation in standard form.ANOTHER WAYx2 8x 16You can also solve theequation in Example 2by factoring.x2Write original equation.x2 8x 16 0 8x 16 0(x 4)(x 4) 0So, x 4.Add 16 to each side.Step 2 Graph the related functiony x2 8x 16.6Step 3 Find the x-intercept. The onlyx-intercept is at the vertex, (4, 0).4y2So, the solution is x 4.y x 2 8x 16246 xSolving a Quadratic Equation: No Real SolutionsSolve x2 2x 4 by graphing.SOLUTION6Method 1yWrite the equation in standard form, x2 2x 4 0. Then graph therelated function y x2 2x 4, as shown at the left.There are no x-intercepts. So, x2 2x 4 has no real solutions.4Method 22Graph each side of the equation.y x 2 2x 4 4 22 xy x2Left sidey 2x 4Right sidey 2x 4The graphs do not intersect.So, x2 2x 4 has no real solutions.Monitoring Progressy42y x 22x 2Help in English and Spanish at BigIdeasMath.comSolve the equation by graphing.4. x2 36 12x5. x2 4x 06. x2 10x 257. x2 3x 38. x2 7x 69. 2x 5 x2Concept SummaryNumber of Solutions of a Quadratic EquationA quadratic equation has: two real solutions when the graph of its related function has two x-intercepts. one real solution when the graph of its related function has one x-intercept. no real solutions when the graph of its related function has no x-intercepts.Section 9.2hsnb alg1 pe 0902.indd 491Solving Quadratic Equations by Graphing4912/5/15 8:57 AM
Finding Zeros of FunctionsRecall that a zero of a function is an x-intercept of the graph of the function.Finding the Zeros of a FunctionThe graph of f (x) (x 3)(x2 x 2) is shown. Find the zeros of f.y6SOLUTIONThe x-intercepts are 1, 2, and 3.So, the zeros of fare 1, 2, and 3.2 44 22Check 3 2) 0 f (2) (2 3)(22 2 2) 04 x1 f ( 1) ( 1 3)[( 1)2 ( 1) 2] 0f (3) (3 f(x) (x 3)(x 2 x 2)3)(32The zeros of a function are not necessarily integers. To approximate zeros, analyze thesigns of function values. When two function values have different signs, a zero liesbetween the x-values that correspond to the function values.Approximating the Zeros of a FunctionThe graph of f (x) x2 4x 1 is shown.Approximate the zeros of f to the nearest tenth.y2SOLUTION 4There are two x-intercepts: one between 4 and 3,and another between 1 and 0. 33Make tables using x-values between 4 and 3, andbetween 1 and 0. Use an increment of 0.1. Look fora change in the signs of the function values. 3.6 3.51 x 2 3.4f(x) x 2 4x 1x 3.9 3.8 3.7 3.3 3.2 3.1f (x)0.610.24 0.11 0.44 0.75 1.04 1.31 1.56 1.79change in signsANOTHER WAYYou could approximateone zero using a tableand then use the axisof symmetry to findthe other zero.x 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1f (x) 1.79 1.56 1.31 1.04 0.75 0.44 0.110.240.61The function values that are closest to 0 correspondto x-values that best approximate the zeros of the function.change in signsIn each table, the function value closest to 0 is 0.11. So, the zeros of fare about 3.7 and 0.3.Monitoring ProgressHelp in English and Spanish at BigIdeasMath.com10. Graph f (x) x2 x 6. Find the zeros of f.11. Graph f (x) x2 2x 2. Approximate the zeros of f to the nearest tenth.492Chapter 9hsnb alg1 pe 0902.indd 492Solving Quadratic Equations2/5/15 8:57 AM
Solving Real-Life ProblemsReal-Life ApplicationA football player kicks a football 2 feet above the ground with an initial verticalvelocity of 75 feet per second. The function h 16t2 75t 2 represents theheight h (in feet) of the football after t seconds. (a) Find the height of the football eachsecond after it is kicked. (b) Use the results of part (a) to estimate when the height ofthe football is 50 feet. (c) Using a graph, after how many seconds is the football50 feet above the ground?SOLUTIONSeconds, tHeight, h021612883834465 23a. Make a table of values starting with t 0 seconds using an incrementof 1. Continue the table until a function value is negative.The height of the football is 61 feet after 1 second, 88 feet after 2 seconds,83 feet after 3 seconds, and 46 feet after 4 seconds.b. From part (a), you can estimate that the height of the football is 50 feet between0 and 1 second and between 3 and 4 seconds.Based on the function values, it is reasonable to estimate that the height of thefootball is 50 feet slightly less than 1 second and slightly less than 4 secondsafter it is kicked.c. To determine when the football is 50 feet above the ground, find the t-values forwhich h 50. So, solve the equation 16t2 75t 2 50 by graphing.Step 1 Write the equation in standard form. 16t2 75t 2 50Write the equation. 16t2 75t 48 0REMEMBEREquations have solutions,or roots. Graphs havex-intercepts. Functionshave zeros.Subtract 50 from each side.Step 2 Use a graphing calculator tograph the related functionh 16t2 75t 48.50h 16t 2 75t 48 16 10Step 3 Use the zero feature to findthe zeros of the function.50 1 ZeroX .76477436 10506Y 0 1 ZeroX 3.9227256 106Y 0The football is 50 feet above the ground after about 0.8 second and about3.9 seconds, which supports the estimates in part (b).Monitoring ProgressHelp in English and Spanish at BigIdeasMath.com12. WHAT IF? After how many seconds is the football 65 feet above the ground?Section 9.2hsnb alg1 pe 0902.indd 493Solving Qu
9.1 Properties of Radicals 9.2 Solving Quadratic Equations by Graphing 9.3 Solving Quadratic Equations Using Square Roots 9.4 Solving Quadratic Equations by Completing the Square 9.5 Solving Quadratic Equations Using the Quadratic Formula 9.6 Solving Nonlinear Systems of Equations 9 Solving Quadratic Equations
Lesson 2a. Solving Quadratic Equations by Extracting Square Roots Lesson 2b. Solving Quadratic Equations by Factoring Lesson 2c. Solving Quadratic Equations by Completing the Square Lesson 2d. Solving Quadratic Equations by Using the Quadratic Formula What I Know This part will assess your prior knowledge of solving quadratic equations
6.3 – Solving Quadratic-Quadratic Systems of Equations 2. Quadratic‐quadratic system– A system of equations involving two quadratic equations involving the same variables. A graph of this system involves two _. The solution to a system of equations from a graph is the point(s) - or ordered pair(s) (,)x y - where the
Solve a quadratic equation by using the Quadratic Formula. Learning Target #4: Solving Quadratic Equations Solve a quadratic equation by analyzing the equation and determining the best method for solving. Solve quadratic applications Timeline for Unit 3A Monday Tuesday Wednesday Thursday Friday January 28 th th Day 1- Factoring
SOLVING QUADRATIC EQUATIONS . Unit Overview . In this unit you will find solutions of quadratic equations by completing the square and using the quadratic formula. You will also graph quadratic functions and rewrite quadratic functions in vertex forms. Many connections between algebra and geometry are noted.
Quadratic Equations Reporting Category Equations and Inequalities Topic Solving quadratic equations over the set of complex numbers Primary SOL AII.4b The student will solve, algebraically and graphically, quadratic equations over the set of complex numbers. Graphing calculators will be used for solving and for confirming the algebraic solutions.
Section 9.3: Solving Quadratic Equations by Factoring Section 9.4: The Quadratic Formula Section 9.5: Complex Numbers . Solving Quadratic Equations Mini-Lesson Page 314 Problem 1 MEDIA EXAMPLE - HOW MANY AND WHAT KIND OF SOLUTIONS? Use your graphing calculator to help you determine the number and type of solutions to each of the quadratic .
Lesson 1: Using the Quadratic Formula to Solve Quadratic Equations In this lesson you will learn how to use the Quadratic Formula to find solutions for quadratic equations. The Quadratic Formula is a classic algebraic method that expresses the relation-ship between a qu
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