Section 1.5 – Solve Quadratic Equations - McGraw-Hill Ryerson

Example 1Select a Strategy to Solve a Quadratic Equationa) Solve 2x 2 12x 14 5 0 byi) completing the squareii) using a graphing calculatoriii) factoringiv) using the quadratic formulab) Which strategy do you prefer? Justify your reasoning.Solutiona) i) 2x 2 12x 14 5 0x 2 6x 7 5 0x 2 6x 9 9 7 5 0(x 3) 2 16 5 0(x 3) 2 5 16x 3 5 4 or x 3 5 4The solutions are x 5 7 and x 5 1.ii) • Use the window settings shown.• Graph Y1 5 2x 2 12x 14.Divide both sides by 2.Take the square root of both sides.Technology TipRefer to the UseTechnology feature onpage 33 to see how tofind zeros using aTI-Nspire TM CASgraphing calculator.• Use the Zero operation to find the x-intercepts.The solutions are x 5 1 and x 5 7.iii) 2x 2 12x 14 5 0x 2 6x 7 5 0(x 7)(x 1) 5 0x 7 5 0 or x 1 5 0x 5 7 or x 5 1Divide both sides by 2.Find the binomial factors of the trinomialx 2 — 6x — 7.44 MHR • Functions 11 • Chapter 1

iv) 2x 2 12x 14 5 0x 2 6x 7 5 0a 5 1, b 5 6, ________ and c 5 7.x 5 ​ b ____ ​ b 2 4ac ​2a________________5 ​ (6) ______ ​ (6) 2 4(1)(7) ​___ 2(1)5 ​__6 ​ 64 ​25 ​ __ 6 82 ​5 ​ 14_ ​or ​2_2 2 ​5 7 or 1Divide both sides by 2.Substitute the values of a, b, and c into thequadratic formula and simplify.b) While all four methods produce the same solutions, factoring isprobably the best strategy for this example. The quadratic expressionis easy to factor, so this method is the fastest. If the quadraticexpression could not be factored, either the graphing calculatormethod or using the quadratic formula would be preferred.ConnectionsIn this example, theroots are integers.However, manyquadratic equationshave irrational roots.If exact roots areasked for, then eithercompleting the squareor the quadratic formulais a better methodto use. The graphingcalculator methodwill only provideapproximations.Solving 2x 2 12x 14 5 0 is equivalent to finding the zeros, orx-intercepts, of the function f (x) 5 2x 2 12x 14. The two solutionsin Example 1 represent the two x-intercepts of the functionf (x) 5 2x 2 12x 14. However, not all quadratic functions havetwo x-intercepts. Some have one x-intercept, while others have nox-intercepts. The next example illustrates this.Example 2Connect the Number of Zeros to a GraphFor each quadratic equation given in the form ax 2 bx c 5 0, graphthe related function f (x) 5 ax 2 bx c using a graphing calculator.State the number of solutions of the original equation. Justify eachanswer.a) 2x 2 8x 5 5 0b) 8x 2 11x 5 5 0c) 4x 2 12x 9 5 0Solutiona) The parabola opens downward and the vertexis located above the x-axis, so the function hastwo zeros.The equation 2x 2 8x 5 5 0 has twosolutions.1.5 Solve Quadratic Equations • MHR 45

) The parabola opens upward and the vertex islocated above the x-axis, so the function has nozeros.The equation 8x 2 11x 5 5 0 has no realsolutions.c) The parabola opens downward and the vertexis located on the x-axis. This function has onezero.The equation 4x 2 12x 9 5 0 has onesolution.The graph of a quadratic function gives you a visual understanding ofthe number of x-intercepts. Without a graphing calculator, it can bequite time-consuming to create this visualization. Is there a way that thenumber of zeros can be identified without drawing a graph? The nextexample revisits Example 2 using the quadratic formula to see if a patterncan be identified that will tell the number of zeros without graphing.Example 3ConnectionsEngineers use the zerosof a quadratic functionto help mathematicallymodel the supportstructure needed for abridge that must span agiven distance.Connect the Number of Zeros to the Quadratic FormulaSolve each quadratic equation in Example 2 using the quadratic formula.Give answers for the x-intercepts as exact values. Compare the resultswith the conclusion for the number of x-intercepts found in Example 2.Solutiona) 2x 2 8x 5 5 0a 5 2, b 5 8, and c 5 5.________x 5 ​ b ____ ​ b 2 4ac ​2a_______________5 ​ 8 _____ ​ 8 2 4(2)(5) ​​2(2)___5 ​___8 ​ 24 ​4 ____x 5 ​___8 2​ 6 ​​or x 5 ​___8 2​ 6 ​44____x 5 ​__4 ​ 6 ​ or x 5 ​__4 ​ 6 ​22The answer of two solutions from Example 2 is verified by thequadratic formula. There are two solutions because the valueunder the radical sign is positive, so it can be evaluated to give twoapproximate roots.46 MHR • Functions 11 • Chapter 1

) 8x 2 11x 5 5 0a 5 8, b 5 11, and c 5 5.________x 5 ​ b ____ ​ b 2 4ac ​2a _______________5 ​ (11) ______ ​ (11) 2 4(8)(5) ​_____ 2(8)5 ​___11 ​ 39 ​16Since the square root of a negative value is not a real number, there isno real solution to the quadratic equation.c) 4x 2 12x 9 5 0a 5 4, b 5 12, and c 5 9.________x 5 ​ b ____ ​ b 2 4ac ​2a________________5 ​ 12 _____ ​ 12 2 4(4)(9) ​​__2(4)5 ​___12 ​ 0 ​85 ​ _ 128 ​5 ​ 3_2 ​There is one solution because the value under the square root is zero.This means that there is exactly one root to the equation4x 2 12x 9 5 0.Example 3 shows that the value under the radical sign in the quadraticformula determines the number of solutions for a quadratic equation andthe number of zeros for the related quadratic function.Example 4Use the Discriminant to Determine the Number of SolutionsFor each quadratic equation, use the discriminant to determine thenumber of solutions.a) 2x 2 3x 8 5 0 b) 3x 2 5x 11 5 0 c) ​ 1_4 ​x2 3x 9 5 0Solutiona) 2x 2 3x 8 5 0a 5 2, b 5 3, and c 5 8.b 2 4ac 5 3 2 4(2)(8)5 9 645 73discriminant• the expressionb 2 — 4ac, the value ofwhich can be used todetermine the numberof solutions to aquadratic equationax 2 + bx + c = 0• When b 2 4ac > 0,there are two solutions.• When b 2 4ac 5 0,there is one solution.• When b 2 4ac < 0,there are no solutions.1.5 Solve Quadratic Equations • MHR 47

Since the discriminant is greater than zero,there are two solutions.You can check this result using a graphingcalculator.b) 3x 2 5x 11 5 0a 5 3, b 5 5, and c 5 11.b 2 4ac 5 (5) 2 4(3)(11)5 25 1325 107Since the discriminant is less than zero,there are no solutions.c) ​ 1_4 ​x2 3x 9 5 0a 5 ​ 1_ ​, b 5 3, and c 5 9.4b 2 4ac 5 (3) 2 4​ ( ​ 1_4 ​ ) ​(9)5 9 95 0Since the discriminant is equal to zero,there is one solution.Key ConceptsA quadratic equation can be solved by– completing the square– factoring– using the quadratic formula– graphingThe number of solutions to a quadratic equation and the number of zeros of the relatedquadratic function can be determined using the discriminant.If b 2 4ac 0, there are twosolutions (two distinct realroots).If b 2 4ac 5 0, thereis one solution (twoequal real roots).If b 2 4ac 0, thereare no real solutions.yyy0 x0 x0 x48 MHR • Functions 11 • Chapter 1

Communicate Your UnderstandingC1 Minh has been asked to solve a quadratic equation of the form ax2 bx c 5 0, but he isunclear whether he should factor, complete the square, use the quadratic formula, or use agraphing calculator. What advice would you give him? Explain.C2 While many techniques can be used to solve a quadratic equation of the form ax2 bx 5 0,what is the easiest technique to use? Why?C3Deepi wants to determine how many x-intercepts a quadratic function has. How can she findthe number of x-intercepts for the function without graphing? Justify your reasoning.A PractiseFor help with questions 1 to 3, refer toExample 1.1. Solve each quadratic equation by factoring.a) x 2 2x 3 5 0b) x 2 3x 10 5 0c) 4x 2 36 5 0d) 6x 2 14x 8 5 0e) 15x 2 8x 1 5 0f) 6x 2 19x 10 5 02. Check your answers to question 1 using agraphing calculator or by substituting eachsolution back into the original equation.3. Solve each quadratic equation using thequadratic formula. Give exact answers.a) 2x 2 17x 27 5 0b) 4x 2 3x 8 5 0c) x 2 x 7 5 0d) x 2 6x 4 5 0e) 3x 2 x 11 5 0f) ​ 1_2 ​x2 4x 1 5 0For help with question 4, refer to Example 2.4. Use Technology Use a graphing calculatorto graph a related function to determine thenumber of roots for each quadratic equation.a) 3x 2 4x 5 5 0b) 8x 2 20x 12.5 5 0c) x 2 2x 5 5 0d) ​ 3_4 ​x2 5x 2 5 0For help with question 5, refer to Example 3.5. Determine the exact values of thex-intercepts of each quadratic function.a) f (x) 5 6x 2 3x 2b) f (x) 5 ​ 1_3 ​x2 4x 8c) f (x) 5 ​ 3_4 ​x2 2x 7d) f (x) 5 ​ 1_4 ​x2 2x 4For help with question 6, refer to Example 4.6. Use the discriminant to determine thenumber of roots for each quadraticequation.Ba) x 2 5x 4 5 0b) 3x 2 4x ​ 4_3 ​5 0c) 2x 2 8x 9 5 0d) 2x 2 0.75x 5 5 0Connect and Apply7. Which method would you use to solveeach equation? Justify your choice. Then,solve. Do any of your answers suggest thatyou might have used another method?Explain.a) 2x 2 5x 12 5 0 b) x 2 25 5 0c) 2x 2 3x 1 5 0 d) ​ 1_2 ​x2 4x 5 0e) 3x 2 4x 2 5 0 f) x 2 4x 4 5 0g) 0.57x 2 3.7x 2.5 5 0h) 9x 2 24x 16 5 01.5 Solve Quadratic Equations • MHR 49

8. Determine the value(s) of k for which thequadratic equation x 2 kx 9 5 0 willhavea) two equal real rootsb) two distinct real roots9. a) Create a table ofvalues for thefunctionf (x) 5 2x 2 3xfor the domain{2, 1, 0, 1, 2, 3, 4}.b) Graph this quadratic function.c) On the same set of axes, graph the liney 5 6.d) Use your graph to determine theapproximate x-values where the liney 5 6 intersects the quadratic function.e) Determine the x-values for the points ofintersection of f (x) 5 2x 2 3x and thehorizontal line y 5 6 algebraically.10. Use Technology Check your answer toquestion 9 using a graphing calculator.11. What value(s) of k, where k is an integer,will allow each quadratic equation to besolved by factoring?a) x 2 kx 12 5 0RepresentingConnectingReasoning and ProvingProblem SolvingCommunicatingb) x 2 kx 5 8 c) x 2 3x 5 kSelecting ToolsReflecting12. The height, h, in metres, above the groundof a football t seconds after it is thrown canbe modelled by the functionh(t) 5 4.9t 2 19.6t 2. Determine howlong the football will be in the air, to thenearest tenth of a second.13. A car travelling at v kilometres per hourwill need a stopping distance, d, in metres,without skidding that can be modelledby the function d 5 0.0067v 2 0.15v.Determine the speed at which a car canbe travelling to be able to stop in eachdistance. Round answers to the nearesttenth of a metre.a) 37 m b) 75 m c) 100 m14. A by-law restricts the height of structuresin an area close to an airport. To conformwith this by-law, fuel storage tanks withdifferent capacities are built by varying theradius of the cylindrical tanks. The surfacearea, A, in square metres, of a tank withradius r, in metres, can be approximatelymodelled by the quadratic functionA(r) 5 6.28r 2 47.7r. What is the radius ofa tank with each surface area?a) 1105 m 2b) 896.75 m 215. The length of a rectangle is 2 m more thanthe width. If the area of the rectangle is20 m 2 , what are the dimensions of therectangle, to the nearest tenth of a metre?16. A building measuring 90 m by 60 m is tobe built. A paved area of uniform widthwill surround the building. The paved areais to have an area of 9000 m 2 . How wide isthe paved area?paved area17. If the same length is cut off three pieces ofwood measuring 21 cm, 42 cm, and 45 cm,the three pieces of wood can be assembledinto a right triangle. What length needs tobe cut off each piece?18. In Vancouver, the height, h, in kilometres,that you would need to climb to see to theeast coast of Canada can be modelled bythe equation h 2 12 740h 5 20 000 000.If the positive root of this equation is thesolution, find the height, to the nearestkilometre.50 MHR • Functions 11 • Chapter 1

19. Chapter Problem Andrea has been askedto determine when (if ever) the volume,V, in hundreds of shares, of a company’sstock, which can be modelled by thefunction V(x) 5 250x 5x 2 , after beinglisted on the stock exchange for x weeks,will reacha) 275 000 shares in a weekb) 400 000 shares in a weekWhat answer should Andrea give?20. Small changesto a quadraticRepresentingequation can havelarge effects on theConnectingsolutions. Illustratethis statement bysolving each quadratic equation.a) x 2 50x 624 5 0b) x 2 50x 625 5 0c) x 2 50x 626 5 0Achievement CheckReasoning and ProvingProblem SolvingCommunicatingSelecting ToolsReflecting21. A diver followed a path defined byh(t) 5 4.9t 2 3t 10 in her dive, wheret is the time, in seconds, and h representsher height above the water, in metres.a) At what height did the diver start herdive?b) For how long was the diver in the air?c) The 4.9 in front of the t 2 term isconstant because it relates to theacceleration due to gravity on Earth. Ifthe diver always starts her dives fromthe same height, what other value in thequadratic expression will never change?d) What is the only value in the quadraticexpression that can change? Suggest away in which this value can change.e) If the value in part d) changed to 6, howmuch longer would the diver be in theair?C Extend22. Complete the square on the expressionax 2 bx c 5 0 to show how thequadratic formula is obtained.23. A cubic block of concrete shrinks as itdries. The volume of the dried block is30.3 cm 3 less than the original volume,while the length of each edge hasdecreased by 0.1 cm. Determine the edgelength and volume of the concrete blockbefore it dried.24. In the diagram, the square has side lengthsof 6 m. The square is divided into three righttriangles and one isosceles triangle. Theareas of the three right triangles are equal.a) Find the value of x.b) Find the area of the acute isoscelestriangle.x6 m6 m25. Math Contest If f (x) 5 2x 2 13x c andf (c)5 16, then one possible value for c isA 2 B 2 C 4 D 826. Math Contest The functionf (x) 5 3x 2 9x 3 has x-interceptsp and q. The value of p pq q is___A 2B 3 5​ 13 ​C 0xD 427. Math Contest The squares MNOP andIJKL overlap as shown. K is the centre ofMNOP. What is the area of quadrilateralKROQ in terms of the area of MNOP?MPKQNOJRLI1.5 Solve Quadratic Equations • MHR 51