4-2: Quadratic Equations - Welcome To Mrs. Plank's Class!

On September 8, 1998, Mark McGwire of the St. LouisCardinals broke the home-run record with his 62nd home run of thep li c a tiyear. He went on to hit 70 home runs for the season. Besides hittinghome runs, McGwire also occasionally popped out. Suppose the ball was 3.5 feetabove the ground when he hit it straightup with an initial velocity of 80 feet persecond. The function d(t) 80t 16t 2 3.5 gives the ball’s height above theground in feet as a function of time inseconds. How long did the catcherhave to get into position to catch theball after it was hit? This problem willbe solved in Example 3.BASEBALLonAp Solve quadraticequations. Use thediscriminant todescribe theroots ofquadraticequations.l WorealdOBJECTIVESQuadratic EquationsR4-2A quadratic equation is a polynomial equation with a degree of two. Solvingquadratic equations by graphing usually does not yield exact answers. Also,some quadratic expressions are not factorable over the integers. Therefore,alternative strategies for solving these equations are needed. One suchalternative is solving quadratic equations by completing the square.Completing the square is a useful method when the quadratic is not easilyfactorable. It can be used to solve any quadratic equation. Remember that, forany number b, the square of the binomial x b has the form x2 2bx b2. Whencompleting the square, you know the first term and middle term and need tosupply the last term. This term equals the square of half the coefficient of the1middle term. For example, to complete the square of x2 8x, find (8) and2square the result. So, the third term is 16, and the expression becomesx2 8x 16. Note that this technique works only if the coefficient of x2 is 1.Example1 Solve x2 6x 16 0.This equation can be solved by graphing, factoring, or completing the square.Method 1Solve the equation by graphing the relatedfunction f(x) x2 6x 16. The zeros ofthe function appear to be 2 and 8.Method 2Solve the equation by factoring.x2 6x 16 0(x 2)(x 8) 0 Factor.x 2 0orx 8 0x 2x 8The roots of the equation are 2 and 8.[ 10, 10] scl:1 by [ 30, 10] scl:5Lesson 4-2Quadratic Equations213

Method 3Solve the equation by completing the square.x2 6x 16 0x2 6x 162x 6x 9 16 9(x 3)2 25x 3 5x 3 5x 8Add 16 to each side. 6Complete the square by adding 2Factor the perfect square trinomial.Take the square root of each side. 2or 9 to each side.x 3 5x 2orThe roots of the equation are 8 and 2.Although factoring may be an easier method to solve this particular equation,completing the square can always be used to solve any quadratic equation.When solving a quadratic equation by completing the square, the leadingcoefficient must be 1. When the leading coefficient of a quadratic equation isnot 1, you must first divide each side of the equation by that coefficient beforecompleting the square.Example2 Solve 3n2 7n 7 0 by completing the square.Notice that the graph of the relatedfunction, y 3x2 7x 7, does notcross the x-axis. Therefore, the roots ofthe equation are imaginary numbers.Completing the square can be used tofind the roots of any equation, includingone with no real roots.3n2 7n 7 0[ 10, 10] scl:1 by [ 10, 10] scl:1n277 n 03373Divide each side by 3.7373n2 n 734936Subtract from each side.73 764936 n 76 2 33 56Factor the perfect square trinomial.35 76n i Take the square root of each side.67635 76n i Subtract from each side.635 635 The roots of the equation are i or .76214Chapter 4Polynomial and Rational Functions24936Complete the square by adding or to each side.n2 n 7 i6

Completing the square can be used to develop a general formula for solvingany quadratic equation of the form ax2 bx c 0. This formula is called theQuadratic Formula.QuadraticFormulaThe roots of a quadratic equation of the form ax 2 bx c 0 witha 0 are given by the following formula.2 4acb x 2a b The quadratic formula can be used to solve any quadratic equation. It isusually easier than completing the square.l WoreaAponldRExamplep li c a ti3 BASEBALL Refer to the application at the beginning of the lesson. How longdid the catcher have to get into position to catch the ball after if was hit?The catcher must get intoposition to catch the ballbefore 80t 16t 2 3.5 0.This equation can be writtenas 16t 2 80t 3.5 0.Use the Quadratic Formulato solve this equation.Ball reachesits highestpoint andstarts backdown.f (t )10075Height(feet)50Ball is hit.f (t ) 16t 2 80t 3.525Ball hitsthe ground. 1 O12345tTime (seconds)b2 4ac 2a t b 802 4( 16)(3.5 ) t 80 2( 16)a 16, b 80,and c 3.56624 t 32 80 6624 t 32 80 ort 0.046624 t 32 80 t 5.04The roots of the equation are about 0.04 and 5.04. Since the catcher has apositive amount of time to catch the ball, he will have about 5 seconds to getinto position to catch the ball.In the quadratic formula, the radicand b2 4ac is called the discriminant ofthe equation. The discriminant tells the nature of the roots of a quadraticequation or the zeros of the related quadratic function.Lesson 4-2Quadratic Equations215

DiscriminantNature of Roots/ZerosGraphyb 2 4ac 0two distinct real roots/zerosOxyexactly one real root/zerob 2 4ac 0(The one real root is actuallya double root.)no real roots/zerob 2 4ac0xy(two distinct imaginaryroots/zeros)ExampleOOx4 Find the discriminant of x2 4x 15 0 and describe the nature of theroots of the equation. Then solve the equation by using the QuadraticFormula.The value of the discriminant, b2 4ac, is ( 4)2 4(1)(15) or 44. Since thevalue of the discriminant is less than zero, there are no real roots.2 4acb 2aThe graph of y x2 4x 15verifies that there are no real roots. x b ( 4) 44 2(1) x 4 2i 11 x 2x 2 i 11 The roots are 2 i 11 and 2 i 11 .[ 10, 10] scl:1 by [ 10, 50] scl:511The roots of the equation in Example 4 are the complex numbers 2 i and 2 i 11 . A pair of complex numbers in the form a bi and a bi arecalled conjugates. Imaginary roots of polynomial equations with real coefficientsalways occur in conjugate pairs. Some other examples of complex conjugates arelisted below.i and iComplexConjugatesTheorem216Chapter 4 1 i and 1 i i 2 and i 2 Suppose a and b are real numbers with b 0. If a bi is a root of apolynomial equation with real coefficients, then a bi is also a root of theequation. a bi and a bi are conjugate pairs.Polynomial and Rational Functions

There are four methods used to solve quadratic equations. Twomethods work for any quadratic equation. One method approximates anyreal roots, and one method only works for equations that can be factored overthe integers.Solution MethodGraphingSituationUsually, only approximate solutions are shown.Examples6x 2 x – 2 0If roots are imaginary (discriminant is less thanf (x )zero), the graph has no x-intercepts, and thesolutions must be found by another method.f (x ) 6x 2 x 2xOGraphing is a good method to verify solutions.21x 3 or x 2 x 2 2x 5 0discriminant: ( 2)2 4(1)(5) 16f (x )84f (x ) x 2 2x 5 4 2 O4x2The equation has no real roots.FactoringWhen a, b, and c are integers and theg2 2g 8 0discriminant is a perfect square or zero, thisdiscriminant: 22 4(1)( 8) 36method is useful. It cannot be used if thediscriminant is less than zero.g2 2g 8 0(g 4)(g 2) 0g 4 0org 2 0g 4Completing theThis method works for any quadratic equation.SquareThere is more room for an arithmetic error thanwhen using the Quadratic Formula.g 2r2 4r 6 0r2 4r 6r2 4r 4 6 4(r 2)2 10r 2 10 r 2 QuadraticFormulaThis method works for any quadratic equation.10 2s2 5s 4 0 5 52 4(2)(4 ) s 2(2) 5 7 s 4 5 i 7 s 4Lesson 4-2Quadratic Equations217

Example5 Solve 6x2 x 2 0.Method 2: FactoringFind the discriminant.Method 1: GraphingGraph y 6x2 x 2.b2 4ac 12 4(6)(2) or 47The discrimimant is less thanzero, so factoring cannot beused to solve the equation.[ 10, 10] scl:1 by [ 50, 50] scl:1The graph does not touch the x-axis,so there are no real roots for thisequation. You cannot determine theroots from the graph.Method 4: Quadratic FormulaFor this equation, a 6,b 1, c 2.Method 3: Completing the Square6x2 x 2 01613x2 x 0b2 4ac b x 2a11631111x2 x 614431441 247x 121441 47x i 1212x2 x 1 2(6) 1 47 x 12 1 i 47 x 12The Quadratic Formula worksand requires fewer steps thancompleting the square. 47x i 1122 4 1(6)(2) x 12 1 i 47 x 12Completing the square works, butthis method requires a lot of steps. 1 i 47 .The roots of the equation are 12C HECKCommunicatingMathematicsFORU N D E R S TA N D I N GRead and study the lesson to answer each question.1. Write a short paragraph explaining how to solve t 2 6t 4 0 by completingthe square.2. Discuss which method of solving 5p2 13p 7 0 would be most appropriate.Explain. Then solve.218Chapter 4 Polynomial and Rational Functions

3. Describe the discriminant of the equation represented by each graph.a.b.yOyxxO4. Mathc.yOxJournal Solve x 2 4x 5 0 using each of the four methods discussedin this lesson. Which method do you prefer? Explain.Guided PracticeSolve each equation by completing the square.5. x 2 8x 20 06. 2a2 11a 21 0Find the discriminant of each equation and describe the nature of the roots of theequation. Then solve the equation by using the Quadratic Formula.7. m2 12m 36 08. t 2 6t 13 0Solve each equation.9. p2 6p 5 010. r 2 4r 10 011. ElectricityOn a cold day, a 12-volt car battery has a resistance of0.02 ohms. The power available to start the motor is modeled by the equationP 12 I 0.02 I 2, where I is the current in amperes. What current is needed toproduce 1600 watts of power to start the motor?E XERCISESPracticeSolve each equation by completing the square.A12. z2 2z 24 03115. d 2 d 04813. p2 3p 88 014. x 2 10x 21 016. 3g2 12g 417. t 2 3t 7 018. What value of c makes x 2 x c a perfect square?19. Describe the nature of the roots of the equation 4n2 6n 25. Explain.Find the discriminant of each equation and describe the nature of the roots of theequation. Then solve the equation by using the Quadratic Formula.B20. 6m2 7m 3 021. s2 5s 9 022. 36d 2 84d 49 023. 4x2 2x 9 024. 3p2 4p 825. 2k2 5k 926. What is the conjugate of 7 i 5 ?27. Name the conjugate of 5 2i.www.amc.glencoe.com/self check quizLesson 4-2 Quadratic Equations219