Solving Quadratic Equations
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1. 9x2 1 6x 7 Simplify and then solve each equation A a x x 1 12 D De PO 50 2 2 b 2x x 4 x4 4 e x 2 2 x 2 3x 5 c 3x e 2 2x 4 x f Qx 1 x 2 8 Determine the roots of each equation a x 4x 32 0 d x 5x 14 b x llx 30 0 e 4x 25 20x c 5x 28x 12 0 f 3x 16x 7 5 9 Solve each equation Round your answers to two decimal places a x 5x 2 0 d x x 5 2x 7 b 4x 8x 3 0 e 3x 5x 3 x 4x 1 c x 1 4 2x f x 3 2x 15 320 6 1 Solving Quadratic Equations NEL 6 1 10 Conor has a summer lawn mowing business Based on experience Conor knows that P 5x2 200x 1500 models his profit P in dollars where x is the amount in dollars charged per lawn a How much does he need to charge if he wants to break even b How much does he need to charge if he wants to have a profit of 500 11 Stacey maintains the gardens in the city parks In the summer she plans to build a walkway through the rose garden The area of the walkway A in square metres is given by A 160x 4x2 where x is the width of the walkway in metres If the area of the walkway must be 900 m2 determine the width 12 Patrick owns an apartment building He knows that the money he earns in a month depends on the rent he charges This relationship PAU EOR En COnnorhOn l can be modelled by E R 1650 R where F is Patrick s2. 4 0 2x2 15x 4 A NEL Chapter 6 317 graphed y 2x2 15x 4 using these window settings From the graph could see that one x intercept was between 1 and 0 and the other x intercept was between 7 and 8 Using the Zero operation of the calculator estimated that the x intercepts were about 0 258 and 7 758 Tech Support For help determining the zeros of a relation using a TI 83 84 graphing calculator see Appendix B 8 If you are using a Tl nspire see Appendix B 44 rounded the solutions to two decimal places The solutions are x 0 26 and x 7 76 These are reasonable estimates since the solutions are not exact EXAMPLE 4 Reflecting on the reasonableness of a solution A ball is thrown from the top of a seaside cliff Its height 4 in metres above the sea after seconds can be modelled by 4 527 21 120 How long will the ball take to fall 20 m below its initial height Jacqueline s Solution h 5 21 120 h 5 0 21 0 120 i let t O to determine the initial height of the ball h 120 The cliff is 120 m high so the ball starts 120 m above the sea The initial height of the ball was 120 m When 120 20 100 the ball had fallen 20 m below its initial height it was 100 m above the sea substituted 100 for h in the relation wrote the equation in the form 0 ax bx cso that could solve it by graphing or factoring L
3. correct E What quadratic equation can you use to describe Andy s goal of making a profit of 1200 F How can you use your graph for part D to determine the roots of your equation for part E G How many T shirts must be sold to achieve Andy s goal Reflecting H Why did factoring x 120x 2000 help you determine the break even points I Are the roots of the equation x2 120x 2000 0 also zeros or x intercepts of the relation y x 120x 2000 Explain 314 6 1 Solving Quadratic Equations NEL 6 1 J Why would factoring the left side of x 120x 2000 1200 not help you determine the number of T shirts that Andy has to sell K Explain why it would help you solve the equation in part J if you were to write it as x 120x 2000 1200 0 2 L To solve ax b c you isolate x Why would you not isolate x to solve ax bx 0 APPLY the Math The users manual for Arleen s model rocket says that the equation h 5t 40t models the approximate height in metres of the rocket after seconds When will Arleen s rocket reach a height of 60 m Amir s Solution Selecting a factoring strategy substituted 60 for h because 5 40 60 wanted to calculate the time for the height 60 m subtracted 60 from both sides of the equation to make the 5t 40 60 0 right side equal zero did this so that could determine the zeros of
4. 6 1 YOU WILL NEED e grid paper e ruler e graphing calculator an equation that contains at least one term whose highest degree is 2 for example xt x 2 0 a solution a number that can be substituted for the variable to make the equation a true statement for example x 1 is a root of x x 2 0 since 14 1 2 0 Solving Quadratic Equations Use graphical and algebraic strategies to solve quadratic equations INVESTIGATE the Math Andy and Susie run a custom T shirt business From past experience they know that they can model their expected profit in dollars with the relation P x 120x 2000 where x is the number of T shirts they sell Andy wants to sell enough T shirts to earn 1200 Susie wants to sell just enough T shirts to break even because she wants to close the business How can Andy and Susie determine the number of T shirts they must sell to achieve their goals A Why can you use the quadratic equation x 120x 2000 0 to determine the number of T shirts that must be sold to achieve Susie s goal B Factor the left side of the equation in part A Use the factors to determine the number of T shirts that must be sold to achieve Susie s goal C Use your factors for part B to predict what the graph of the profit relation will look like Sketch the graph based on your prediction D Graph the profit relation using a graphing calculator Was your prediction for part C
5. By photosynthesis green plants 50 remove carbon dioxide from monthly earnings in dollars and R is the amount of rent in dollars the air and produce oxygen he charges each tenant a How much will he earn if he sets the rent at 900 b If Patrick wants to earn at least 13 000 between what two values should he set the rent 13 Determine the points of intersection of the line y 2x 7 and the parabola y 2x 3x 5 14 While hiking along the top of a cliff Harlan knocked a pebble over EY the edge The height in metres of the pebble above the ground after t seconds is modelled by 5t 4t 120 a How long will the pebble take to hit the ground b For how long is the height of the pebble greater than 95 m 15 Is it possible to solve a quadratic equation that is not factorable over the set of integers Explain 16 a Describe when and why you would rewrite a quadratic equation to solve it In your answer include x2 2x 15 rewritten as x 2x 15 0 b Explain how the relation y x 2x 15 can be used to solve ge De 15 0 Extending 17 Solve the equations x4 9x2 20 Oand x 9x2 20x 0 by first solving the equation x 9x 20 0 18 Will all quadratic equations always have two solutions Explain how you know and support your claim with examples NEL Chapter 6 321
6. et 4 100 100 5r2 214 120 0 547 21 120 100 0 522 214 20 subtracted 100 from both sides of the equation to make the left side equal to 0 a 318 6 1 Solving Quadratic Equations NEL 6 1 0 54 21 20 multiplied all the terms on both sides of the 0 57 25t 4t 20 equation by 1 because wanted 5t to be 0 5e 5 47 5 positive factored the right side of the equation O 5 4 5 using decomposition stas or 7 5 O 5t 4 t 5 set each factor equal to zero and solved for t 4 p n 5 Since the ball was thrown at t O I knew that the The ball will take 5 s to fall 20 m below solution t 2 didn t make sense used the its initial height l San a t solution t 5 since this did make sense In Summary Key Ideas e A quadratic equation is any equation that contains a polynomial in one variable whose degree is 2 for example x2 6x 9 0 e All quadratic equations can be expressed in the form ax bx c 0 using algebraic strategies In this form the equation can be solved by e factoring the quadratic expression setting each factor equal to zero and solving the resulting equations or e graphing the corresponding relation y ax bx cand determining the zeros or x intercepts Need to Know e Roots and solutions have the same meaning These are all values that satisfy an equation e Some quadratic equatio
7. ns can be solved by factoring Other quadratic equations must be solved by using a graph e f you use factoring to solve a quadratic equation write the equation in the form ax bx c 0 before you try to factor e To solve ax bx c d using a graph graph y ax bx c and y don the same axes The solutions to the equation are the x coordinates of the points where the parabola and the horizontal line intersect CHECK Your Understanding 1 The solutions to each equation are the x intercepts of the corresponding quadratic relation State the quadratic relation a x 4x 4 0 b 2x 9x 5 NEL Chapter 6 319 2 Use the graph of each quadratic relation to determine the roots to each quadratic equation where y 0 b 3 Solve each equation a x x 4 0 d 3x 8 4 0 b x 10 8 0 e x7 5x 6 0 c x 5 0 f x 2x 8 PRACTISING 4 Determine whether the given value is a root of the equation a x 2 x x 6 0 d x gt 8x 10x 3 0 b x 4 x 7x 8 0 e x 5 x 4x 5 0 1 c x 7 2x llx 5 0 f x 2 3x2 2x 8 0 5 Solve each equation by factoring Use an equivalent equation if necessary a x 2x 15 0 d x 5x 0 b x 5x 24 0 e x 6x 16 c x 4x 4 0 f x 12 7x 6 Solve by factoring Verify your solutions aj Oe oe 2 d Gr 42 2 0 b 2x 3x 2 0 e 4x 4x 3 c 3x 4x 15 0 f
8. r 6 s 316 6 1 Solving Quadratic Equations NEL 6 1 Determine the roots of 6x2 11x 10 0 Annette s Solution be Tie 10 0 Since the trinomial in the equation contains no Podu 260 Sum 14 common factors and is one where a 1 used decomposition looked for two numbers whose i aon i F es sum is 11 and whose product is 6 10 60 3 20 3 20 17 4 15 4 15 llv Since the numbers were 15 and 4 used these to decompose the middle term factored the first two terms and then the last two terms Then 6x 15x 4x 10 0 3x 2x 5 2 2x 5 0 2x 5 3x 2 0 divided out the common factor of 2x 5 Ix 5 0 or 3x t2 0 set each factor equal to zero and solved Ix 5 3 2 each equation 5 x 2 3 The roots of 6x 11x 10 0 2 are x 2 and x pi 3 Determine all the values of x that satisfy the equation x 4 3x x 5 If necessary round your answers to two decimal places Karl s Solution decided to write an equivalent equation in the form ax bx c 0 which could solve by graphing or factoring expanded the expression on the right side of the equation x 4 3x x 5 x 4 3x 15x used inverse operations to make the left side of the equation equal to zero couldn t factor the right side of the equation so decided to use a graph 0 3x2 x2 15x
9. the corresponding relation 5 t2 8t 12 0 divided out the common factor 5 2 t 6 0 of 5 Then factored the 2 O0ortr 6 0 trinomial The trinomial will A 5 mee equal zero if either factor equals zero set each factor equal to The rocket is 60 m above the ground zero and solved both equations at 2 s on the way up and 6 s on the This gave me the zeros of the way down parabola NEL Chapter 6 315 Tech Support For help locating the zeros of a relation using a TI 83 84 graphing calculator see Appendix B 8 If you are using a Tl nspire see Appendix B 44 Tech Support For help determining points of intersection using a Tl 83 84 graphing calculator see Appendix B 11 If you are using a Tl nspire see Appendix B 47 verified my solutions by graphing y 5x2 40x 60 and then locating the zeros L My solutions were correct Alex s Solution Selecting a graphing strategy Ha ERRANS Using a graphing calculator YAGA entered the height equation in 3 Y1 substituted the variable y for h and the variable x for t entered 60 in Y2 to determine when the rocket will reach 60 m iq estimated that the rocket will travel about 100 m and be in the air for about 10 s so used these window settings used the Intersect operation to locate the intersection points of the two graphs The rocket is 60 m off the ground after 2 s and afte
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