Unit III Quadratic Equations 1 Section 7.1 Solving Quadratic Equations by Graphing Goal: Solving Quadratic Equations by Graphing Investigating Solutions to Quadratic Equations Eample: A missile fired from ground level is modeled by the quadratic function h(t) = 0t + 10t, where h(t) represents height in meters and t is the time in seconds. h(t) t (a) Indicate the positions along the quadratic trajectory above, where the rocket attains a height of 0? (b) For the function above, h(t) = 0t + 10t which variable is replaced by the height 0 for those positions? (c) Write the resulting quadratic when the variable mentioned above is replaced by 0.
Unit III Quadratic Equations What is a Quadratic Equation? A second degree polynomial equation. Standard Form of a quadratic Equation a + b + c = 0. (d) Use graphing software (https://www.desmos.com/) to determine the times below where the rocket attains a height of 0. Remember: Zeros of a Quadratic Function h(t) y h(t) = 0t + 10t Zeros of a Quadratic Function t The zero(s) of a quadratic function represent the position(s) where the height is The zero(s) of a quadratic function are also referred to as the Solutions to Quadratic Equation Solutions to the quadratic equation 0t + 10t = 0 are the of the quadratic function times where height is 0 are: t1 = and t =
Unit III Quadratic Equations 3 (e) Use the graph below to determine the times where the rocket attained a height of 160 m. 00 180 160 140 10 100 80 60 40 0 h(t) h(t) = t + 10t 1 3 4 5 6 7 8 t t 1 = t = (f) Use the quadratic function h(t) = 0t + 10t to write the quadratic equation that represents the height at 160 m. How can we algebraically attain the time(s) to solve a quadratic equation a + b + c = 0 such as:?
Unit III Quadratic Equations 4 Section 7. Solving Quadratic Equations by Factoring Goal: Solving Quadratic Equations by Factoring (I) Solving Quadratic Equations by Factoring Eample: A missile fired from ground level is modeled by the quadratic function h(t) = 0t + 10t, where h(t) represents height in meters and t is the time in seconds. At a height of 160 m the quadratic function h(t) = 0t + 10t develops into a quadratic equation: 160 = 0t + 10t 00 180 160 140 10 100 80 60 40 0 h(t) t We can determine the two times where the missile attains a height of 160 m by FACTORING the quadratic equation 160 = 0t + 10t and using the zero product property to isolate and solve for t.
Unit III Quadratic Equations 5 Solve for t by factoring: 160 = 0t + 10t Epress in the form a + b + c = 0 Remove the greatest common factor. Factor the remaining trinomial. To isolate t, apply the Zero Product Property If the product of two real numbers is zero ( a b = 0) then one or both must be zero. In other words: a = 0 and b = 0 (II) Review of Factoring (A) Factoring by removing a common factor. Eample: Solve by factoring. (i) 0 + 10 = 0 (ii) 1 = 8
Unit III Quadratic Equations 6 (II) Review of Factoring (B) Solving quadratic equations by the difference of two squares. Perfect Square Numbers or Factors to remember: = 4 3 = 9 4 = 16 5 = 5 6 = 36 7 = 49 8 = 64 9 = 81 10 = 100 11 = 11 1 = 144 13 = 169 14 = 196 15 = 5 Perfect Square Numbers or Factors REMEMBER: The difference of two squares factors by pattern a b = (a b)(a + b) Eample: Solve by factoring. (i) 81 = 49 (ii) 11 4 = 0
Unit III Quadratic Equations 7 (II) Review of Factoring (C) Solving quadratic equations of the form + b + c = 0. Eample: Solve by factoring. (i) + 3 = 18 (ii) p 0p + 48 = 0 (D) Solving quadratic equations of the form a + b + c = 0. Eample: Solve by factoring. (i) 6 11 = 10 (ii) 8 = 15 P.411 #1 #4
Unit III Quadratic Equations 8 Section 7. Solving Quadratic Equations by Factoring (Day ) Goal: Solving Quadratic Equations by Factoring (I) Determining Roots of a Quadratic Equation a + b + c = 0 The roots of a quadratic equation represent the TWO values that are solutions to a + b + c = 0. Eample: Determine the ROOTS for: (a) (3 + 7) = 6 Epress each quadratic equation in standard form a + b + c = 0 and factor. Set each factor equal to zero and solve each linear equation. (b) 3. + 6.4 = 5.6 (c) 1 3 = + 6
Unit III Quadratic Equations 9 (VI) Applications of Quadratic Equation a + b + c = 0 Eample: The path of a missile shot into the air from a ship is modeled by the quadratic function y= 4.9 + 39. + 44.1 where y represents the height in meters and is time in seconds. y Quadratic Path is modeled by: y = 4.9 + 39. + 44.1 Algebraically determine: (a) How long will it take for the missile to hit the ocean? (b) How long will it take for the missile to attain a height of 1.5 metres?
Unit III Quadratic Equations 10 Eample: An osprey dives toward the water to catch a salmon. Its height above the water, in metres, t seconds after it begins its 10 0 30 40 50 dive, is approimated by h( t) 5t 30t 45. Algebraically determine the time it takes for the osprey to reach a return height of 0 m. Height 50 40 30 0 10 time Eample: A travel agency has 16 people signed up for a trip. The revenue for the trip is modeled by the function R() = 100 + 800 + 38400 where represents the number of additional people to sign up. How many additional people must sign up for the revenue to reach $40000? P.411 413 #6, #8, #13, #14
Unit III Quadratic Equations 11 Section 7. Developing Quadratic Functions/Equations (Day 3) Goal: Developing Quadratic Functions/Equations (I) Review of zeros and roots Determining the zeros of a quadratic function f() = a + b + c Procedure: set y = 0 OR f() = 0 solve a + b + c = 0 by factoring Eample: Determine the zeros for f() = 4 1 Determining Roots of a Quadratic Equation a + b + c = 0 Procedure: arrange the quadratic equation to the form a + b + c = 0 solve a + b + c = 0 by factoring/quadratic formula or isolating the variable Eample: Determine the roots for: 5 + 7 6 = 0
Unit III Quadratic Equations 1 (II) Developing a quadratic equation/function given roots/zeros Eample: Determine the quadratic function y = a + b + c that has zeros and 6. How do we reverse the procedure for attaining zeros so we can develop the function? (i) (ii) (iii) Eample: Determine the quadratic equation a + b +c = 0 that has roots and 3 5.
Unit III Quadratic Equations 13 Eample: Determine the quadratic equation that has roots: (a) 3 4 and 3 (b) ± 5 (c) 3 ± 4 How will this be evaluated? Potential multiple choice questions: 1. Which quadratic function has zeros (A) y 4? 1. (B) y (C) (D) y y. Which equation has roots 3 and. 4 (A) (B) (C) (D) 4 11 6 0 4 5 6 0 4 5 6 0 4 11 6 0
Unit III Quadratic Equations 14 Practice Problems: 1. Given the zeros, determine the quadratic function. (a) 3 5 and 1 (b) 1 ± 3 (c) 0 and 4 3 (d) ±4 7. Given the roots, determine the quadratic equation. (a) 7 and 5 (b) ± 3 Answers: 1.(a) y = 10 + 3 (b) y = 11 (c) y = 3 4 (d) y = 11.(a) y = 7 37 + 10 (b) y = 3
Unit III Quadratic Equations 15 Quiz Review: Quadratic Equations 1. Factor the following. (a) + 5 (b) 9 1 (c) + 6 + 8 (d) 3 7 + (e) 5 + 15 0. Determine the intercepts of the following graphs. (a) y 1-8 - 7-6 - 5-4 - 3 - - 1-1 1 (b) y 8 6 4 - - 3-4 - 5 - - 4 6 8-6 -4 3. A missile fired from ground level is modeled by the quadratic function h(t) = 0t + 10t, where h(t) represents height in meters and t is the time in seconds. (a) Write the quadratic equation for the height of the missile at 160 m. (b) At what times does the missile reach a height of 160 m? (c) Write the quadratic equation for the height of the missile at 100 m. (d) At what times does the missile reach a height of 100 m? 00 180 160 140 10 100 80 60 40 0 h(t) 1 3 4 5 6 7 8 t
Unit III Quadratic Equations 16 4. Use factoring to determine the zeros of the following quadratic functions. (a) y = 3 (b) y = + 8 (c) y = 16 (d) y = 4 9 (e) y = 3 10 (f) y = 10 + 4 (g) y = + 9 + 10 (h) y = 6 5 6 (i) y = 50 = 0 (j) y = 3 + 3 36 5. Use factoring to determine the roots of the following equations. (a) ( + 4 ) = 1 (b) ( + 1 ) = 5 (c) 0.5 = 0.5 + 0.75 (d) 1.4 + 8.4 7 = 0 (e) 4. + 6.3 =.1 (f) 1 = 18 (g) 4 3 + 1 3 = 0 (h) 1 4 + = 4 6. The path of a missile shot into the air from a ship is modeled by the quadratic function h(t) = 4.9t + 9.4t + 78.4 where h represents the height in meters and t is time in seconds. (a) Determine how long it takes for the missile to hit the ocean. (b) Determine at what times the missile reaches a height of 10.9 m. 7. An osprey dives toward the water to catch a salmon. Its height above the water, in metres, t seconds after it begins its dive, is approimated by h(t) = 5t 5t + 45. Algebraically determine the time it takes for the osprey to catch the salmon and then reach a return height of 5 m. 8. Travel World Agency has 5 people signed up for a vacation. The revenue for the vacation is modeled by the function R(p) = 50p +400p + 500 where R(p) represents the revenue and p represents the number of additional people to sign up. How many additional people must sign up for the revenue to reach $1500? 9. Determine the quadratic function, y = a + b + c, that has the given zeros of a function. (a) 4 and (b) 0 and 3 1 (c) 3 (d) 3 and 5
Unit III Quadratic Equations 17 10. Determine the quadratic equation, a + b + c = 0, that has the given roots. (a) 3 and 1 (b) 1 3 and 4 3 (c) 3 (d) and 11. Which function has zeros? 1. Which equation has roots 3 and 3? (A) (A) 3 7 6 = 0 (B) y (B) 3 + 7 6 = 0 (C) (C) 3 + 7 + 6 = 0 (D) (D) 3 7 + 6 = 0 y y y 4 Answers 1(a) ( + 5 ) (b) ( 3 + 1 )( 3 1 ) (c) ( 3 1 )( ) (d) (5)( + 4 )( 1 ) (a) = 5, 1 (b) =, 6 3(a) 160 = 0t + 10t (b) sec, 4 sec (b) 100 = 0t + 10t (d) 1 sec, 5 sec 4(a) = 0, 3 (b) = 4, 0 (c) = 4, 4 (d) = (f) = 4, 6 (g) = 5, (h) = 3, 3 3, 3 (e) =, 5 (i) = 5, 5 (j) = 4, 3 1 1 5(a) = 6, (b) =, (c) = 1, 3 (d) = 1, 5 (e) = 1, (f) = 6, 6 (g) = 1, 4 1 (h) = 4 6(a) t = 8 sec (b) t = 1 sec, 5 sec 7. t = 4 sec 8. 10 people 9(a) y = + 8 (b) y = 3 (c) y = 18 (d) y = 5 14 3 10(a) + 3 = 0 (b) 8 3 = 0 (c) 3 = 0 (d) + 6 = 0 11. D 1. B
Unit III Quadratic Equations 18 Section 7.3 Solving Quadratic Equations by Quadratic Formula Goal: Applying the Quadratic Formula to Determine the Roots of a Quadratic Equation (I) Applying the quadratic formula to solve a quadratic equation Eample: A missile fired from ground level is modeled by the quadratic function h(t) = 0t + 10t, where h(t) represents height in meters and t is the time in seconds. 00 180 160 140 10 100 80 60 40 0 h(t) h(t) = 0t + 10t 1 3 4 5 6 7 8 t (a) Write the quadratic equation that could be used to determine the times where the missile attains a height of 80 m. (b) Epress the quadratic equation from (a) above in standard form. (c) Can we solve the quadratic equation from (b) by factoring?
Unit III Quadratic Equations 19 Solving Quadratic Equations by Quadratic Formula: We can solve quadratic equations in the form a + b + c = 0 that do not factor by applying the quadratic formula. = b± b 4ac a Apply the quadratic formula to solve the quadratic equation to determine the times when the missile attains a height of 80m.
Unit III Quadratic Equations 0 (II) Determining EXACT ROOTS of a Quadratic Equation by applying Quadratic Formula What is an EXACT ROOT? Eample: Use a calculator to approimate Eample: Simplify the EXACT ROOT. (a) 80 (b) 7 Eample: Determine the EXACT ROOTS for: (a) = 17 (b) 4( 3) = 7
Unit III Quadratic Equations 1 (III) Solving Quadratic Equations that factor by applying the Quadratic Formula Eample: Solve by factoring 3( + ) = 4( ) Eample: Solve by applying the quadratic formula 3( + ) = 4( ) P.419 40 #a, d #6
Unit III Quadratic Equations Section 7.3 Solving Quadratic Equations by Quadratic Formula (Day ) Goal: Applying the Quadratic Formula to Solve Quadratic Equations and Distinguishing the type of Attainable Roots (I) Investigating the type of attainable roots by application of the quadratic formula. Solve each equation by applying the quadratic formula. (A) + 8 = 0 (B) + 4 = 0 Investigating Attainable Roots (i) What is the value under the square root in (A)? Is it a perfect square? How many roots eist and are they eact or approimate?
Unit III Quadratic Equations 3 Investigating Attainable Roots (ii) What is the value under the square root in (B)? Is it a perfect square? How many roots eist and are they eact or approimate? (iii) What values of b 4ac could lead to approimate answers? (iv) What values of b 4ac could lead to eact answers? (v) Which equation could also be solved by factoring? (vi) What connection can be made between an equation that is factorable and the value of b 4ac?
Unit III Quadratic Equations 4 (II) Investigating the number of zero(s)/root(s) of a quadratic function/equation. A quadratic function/equation can have two, one or no real zeros/roots pending on the value attained underneath the root of the quadratic formula. = b± b 4ac a The value of b 4ac and how it compares to 0 can be used to predict what s happening graphically. The result of b 4ac predicts the number of zeros (-intercepts)/roots Investigate: In each case, determine how the value of b 4ac compares to 0 (>, <, =) to represent what has occurred graphically. (a) Calculate the value of b 4ac and how it compares to 0 (>, <, =) for y = 5 + 4 and sketch the graph. (Check out https://www.desmos.com/) y When b 4ac 0 it means that zero(s)/root(s) are attained.
Unit III Quadratic Equations 5 (b) Calculate the value of b 4ac and how it compares to 0 (>, <, =) for y = + 4 + 4 and sketch the graph. (Check out https://www.desmos.com/) y When b 4ac 0 it means that zero(s)/root(s) are attained. (c) Calculate the value of b 4ac and how it compares to 0 (>, <, =) for y = 4 + 5 and sketch the graph. (Check out https://www.desmos.com/) y When b 4ac 0 it means that zero(s)/root(s) are attained.
Unit III Quadratic Equations 6 (III) Investigating the type of attainable root(s) from a quadratic equation and making a connection to the corresponding graph of the quadratic function. Eample: Apply the quadratic formula to determine the roots of the quadratic equation: ( 7) = 5( + 1) If the value of the radicand (or b 4ac) is in the quadratic formula then there is solution for the quadratic equation. Epress the quadratic equation above (a + b + c = 0) as a quadratic function (y = a y + b + c) and sketch the graph. (Check out https://www.desmos.com/) If the value of the radicand (or b 4ac) is in the quadratic formula then there is zeros for the quadratic function.
Unit III Quadratic Equations 7 Practice Problems: Determine the eact roots for: 1. = 8 5. 3= 0 3. ( 1) 3 = 5 4. 3 =
Unit III Quadratic Equations 8 5. 1 3 + 4 = 15 Answers: 1. 4 ± 6. 1 ± 7 3. 3 ± 3 4. ± 0 6 5. 1 ± 36
Unit III Quadratic Equations 9 Section 7.3 Solving Quadratic Equations by Quadratic Formula (Day 3) Goal: Modeling a Problem with a Quadratic Equation and Solving the Equation (I) Modeling Projectile Motion Eample 1 Quadratic Equation is given that models projectile motion A football is kicked and its trajectory is modeled by the function, h(t) = 4t + 0t + 1 where h(t) represents height in feet ant t is time in seconds. How long does it take for the football to attain a height of 15 feet?
Unit III Quadratic Equations 30 (II) Modeling Revenue Eample: A concert promoter s profit P(s), in dollars, can be modeled by the function P(s) = 8s + 950s 50 where s is the price of a ticket in dollars. (a) If the promoter wants to earn a profit of $0 000, what should be the price of the ticket? (b) Is it possible for the promoter to earn a profit of $30 000? Eplain. P.40 41 #7, #8a, c #10
Unit III Quadratic Equations 31 QUADRATIC EQUATIONS SAMPLE INCLASS ASSIGNMENT QUADRATIC FORMULA: PART A: MULTIPLE CHOICE ( Value: 8 ) Place the letter of the correct response in the space provided to the right. 1. Which quadratic function below has intercepts of 4 and 1 and a y intercept of 4? 1. (A) 6 5 4 3 1 y - - 1-1 1 3 4 5 - - 3-4 - 5-6 (B) - 5-4 - 3 - - 1-1 1 - - 3-4 - 5-6 6 5 4 3 1 y (C) 7 6 5 4 3 1 y (D) 7 6 5 4 3 1 y - - 1-1 1 3 4 5 - - 3-5 - 4-3 - - 1-1 1 - - 3. Which are the zeros of the quadratic function f() = ( 3 4 )( + )?. (A) =, = 4 3 4 (B) =, = 3 (C) =, = 3 4 (D) =, = 3 4
Unit III Quadratic Equations 3 3. Which quadratic equation has roots of 0 and 3? 3. (A) 3 = 0 (B) 3 = 0 (C) 3 + = 0 (D) + 3 = 0 4. What are the roots of the quadratic equation 9 4 = 0? 4. (A) = 3 (B) = 3 (C) = 9 4 (D) = 4 9 5. A golf ball is struck and its trajectory is modeled by the function h(t) = 4t + 0t where h(t) represents height in meters and t is time in seconds. Determine the time it takes for the golf ball to hit the ground after it has been struck. 5. (A) 0 sec (B).5 sec (C) 4 sec (D) 5 sec 6. Which graph represents a quadratic function with no real zeros? 6. y y (A) (B) (C) y (D) y 7. For a quadratic function, the value of b 4ac = 8. Which is true about the graph of the quadratic function? 7. (A) There are three intercepts. (C) There is one intercept. (B) There are two intercepts (D) There are zero intercepts.
Unit III Quadratic Equations 33 8. Mark used the quadratic formula, as shown below, to solve the quadratic equation + 7 1 = 0. He made an error in his calculations. In which step did Mark first make his mistake? 8. (A) Step 1 (B) Step (C) Step 3 (D) Step 4 Step 1: 7 7 4(1)( 1) (1) Step : 7 49 4 Step 3: 7 45 Step 4: 7 3 5 PART B: QUESTIONS ( Value: 16 ) Answer each question in the space provided. Show all your workings to ensure full marks! 9. Solve by factoring. ( 4 ) ( + 4 ) = 1 10. Determine the quadratic equation that has roots of 1 and. ( 4 ) 3
Unit III Quadratic Equations 34 11. Determine the EXACT ROOTS for the quadratic equation: ( 3) = ( + 4) ( 4 ) 1. The trajectory of a missile fired from a ship is modeled by the function h(t) = 3t + 36t + 4 where h represents height in meters and t is time in seconds. Determine the time(s) when the missile reaches a height of 100 m. ( 4 ) ANSWERS: 1. D. C 3. B 4. A 5. D 6. D 7. B 8. B 9. = 6 and = 10. 6 = 0 11. = ± 1. 4 sec and 6 sec
Unit III Quadratic Equations 35 Section 7.3 Solving Quadratic Equations by Quadratic Formula (Day 4) Goal: Modeling a Problem with a Quadratic Equation and Solving the Equation REMEMBER: Area = length width (III) Quadratic Equations Developed through Area Problems Eample 1: Modeling Area Compression Problems The owner of a new home would like to construct a rectangular driveway surrounded by a paving stone border of uniform width on three sides. The dimensions of the driveway including the paving stone border is 5 feet by 1 feet. The pavement in the driveway represents 40 ft. Determine the quadratic equation that models the area of the driveway and use it to determine the width of the paving stone border. 5 ft Driveway 1 ft
30m Unit III Quadratic Equations 36 Eample : Modeling Area Epansion Problems A school playground is rectangular and has a length of 50 m and a width of 30 m as shown. A safety zone of uniform width surrounds the playground. If the entire area is 3500m, what is the width of the safety zone? Safety Zone Playground 50m Eample 3: A rectangular lawn measuring 8 m by 4 m is surrounded by a flower bed of uniform width. The combined area of the lawn and flower bed is 165 m. What is the width of the flower bed?
Unit III Quadratic Equations 37 Practice Problems: 1. A strip of pavement of the same width will be constructed around three sides of 30 m by 0 m Warehouse building. The total area of the building and pavement is 1500 m. Develop the quadratic equation that models the area of the Warehouse and pavement and use it to determine the width of the pavement strip. Warehouse 0 m 30 m. A rectangular picture 7 cm by 5 cm is surrounded by a uniform wooden frame so that the total area of frame and picture is 143 cm. Set up an equation to model this situation and use it to algebraically determine the width of the wooden frame.
Unit III Quadratic Equations 38 3. A rectangular garden, measuring 15 m by 0 m, has a uniform strip removed from the edge of one length and the edge of one width to make a concrete walkway. The area of the remaining garden is 00 m. Develop a quadratic equation that models the area of the new garden and solve it to determine the width of the concrete walkway. 15 m 0 m 4. A carpenter will renovate a 10 ft. by 1 ft. rectangular room by knocking down and moving two walls. A uniform etension out of one length and width wall will double the eisting area. Develop the quadratic equation that models the area of the new room and use it to determine etension of each wall. Answers: 1. 10 m. 3 cm 3. 3.15 m 4. 4.53 ft.
Unit III Quadratic Equations 39 Section 7.4 Applications of Quadratic Equations Goal: Solve Problems that Involve Quadratic Equations (I) Modeling Projectile Motion based on a verbal description Eample: Developing a quadratic function that models projectile motion A baseball is thrown from an initial height of 3 m and reaches a maimum height of 8 m, seconds after it is thrown. What time does the ball hit the ground? Height h(t) time t
Unit III Quadratic Equations 40 (II) Modeling Projectile Motion based on a given function Eample: A diver s path when diving off a platform is given by d = 5t + 10t + 0, where d is the distance above water in feet and t is the time in seconds. (a) How high is the diving platform? (b) After how many seconds is the diver 5 feet above the water? (c) When does the diver enter the water?
Unit III Quadratic Equations 41 (III) Applying Quadratic Equations to Numeracy Problems Consecutive means one number after another so, if the first number is the net number will be +1or 1 Consecutive even (or odd) numbers, you will have and + or You cannot have a negative length or width Eample: Determine two consecutive positive even numbers that have a product of 48. Eample: The sums of squares of two consecutive even integers is 100. Determine the integers.
Unit III Quadratic Equations 4 PRACTICE PROBLEMS: 1. A missile is shot into the air from a ship. The height of the missile above sea level, in metres, t seconds after being shot is approimated by h(t) = 6t + 6t + 4. h(t) t Algebraically determine: (a) the times when the missile attains a height of 1 m. (b) the time when the missile hits the ocean.
Unit III Quadratic Equations 43. An owl perched in a tree ascends to a maimum height of 30 metres at 4 seconds. It spots a rat on the ground and descends to strike it at 10 seconds. (a) Determine the quadratic function that describes the flight path h t (b) Determine the time(s) it takes for the owl to attain a height of 0 m. P.430 P.431 #3, #5, #7, #8
Unit III Quadratic Equations 44 Chapter 7 Quadratic Equations Test Review QUADRATIC FORMULA: b b 4ac a 1. Solve the following quadratic equations using factoring: a) 6 = 3 b) 4 9 = 0 c) 3 = + 10 d) ( + 4) = 10 e) ( + 10) = 4 + 16 f).5 = 80 30 g) 1 4 + 5 = 0. Solve the following using the quadratic formula: a) 4 + 1 = 0 b) + 10 = 0 3. Determine the EXACT roots of the following: a) 4 = 0 b) ( 4) = 3 c) 1 + 14 = 0 d) 3 + 11 = 4 b) e).1 = 18.9 4. Determine the quadratic equation, a + b + c = 0, that has the following roots. a) = 9 = b) = 3 4 = 1 c) = 1 4 = 3 5 d) = ± 5 5. A rocket is fired into the air according to the equation h(t) = t + 4t + 48 where t is the time in minutes and h is the height in meters. (a) Determine the time(s) the rocket is at a height of 3 meters. (b) Determine when the rocket hits the ground. 6. A missile s path when fired from a ship is given by h(t) = 3t + t + 8, where h(t) is the height of the missile in metres and t is the time in seconds. (a) When does the missile hit the water? (b) Approimately when does the missile reach a height of 4m?
Unit III Quadratic Equations 45 7. The revenue made by a drama theater is represented by R = 0 + 80 + 300, where is the number of shows the drama group performs. How many shows should the drama group have to make a profit of $ 000? 8. A rectangular swimming pool has length 30 m and width 0 m. There is a deck of uniform width surrounding the pool. The area of the pool is the same as the area of the deck. Write a quadratic equation that models this situation and use it to determine the width of the deck. 30 m 0 m 9. A driveway has dimensions of 8m 18m. Then a flower bed of uniform width is added to two sides of the driveway. What is the width of the flower bed if the total combined area is 00 m? Driveway 10. Susan decides to build a uniform deck around her pool which has dimensions of 0 m by 10 m. If the total area of the pool and deck measures 300m then write a quadratic equation that models this situation and use it to determine the width of the uniform strip denoted by? 10 m Pool 0 m
Unit III Quadratic Equations 46 11. A rectangular garden, measuring 0 m by 15 m, has a uniform strip removed from the edge of one length and the edge of one width to make a concrete walkway. If the area of the remaining garden is 04 m, what will be the width of the concrete walkway? 15 m 0 m 1(a) Which graph represents a quadratic function with two unequal, real zeros? (b) Which graph represents a quadratic function with two equal, real zeros? (c) Which graph represents a quadratic function with two unequal, unreal zeros? y y (A) (B) (C) y (D) y 13. Determine the number of roots and type of roots for each of the following using b 4ac. a) 3 + 1 = 0 b) + 6 + 9 = 0 c) + 5 3 = 0 14. The product of two odd consecutive integers is 63. Determine the integers.
Unit III Quadratic Equations 47 15. The sum of the squares of two even consecutive integers is 5. Determine the integers. ANSWERS 1(a) 1 0, (b) 3 5 3 (c), (d) 5, 1 (e) 8, (f) 4, 8 (g) 10, (a) 3 (b) 5 3 3 3(a) 1 5 (b) 1, 3 (c) 3 (d) 1 4, 3 3(e) 3 4(a) 11 18 0 (b) 4 3 0 (c) 0 7 3 0 (d) 5 0 5(a) 4 sec (b) 6 sec 6(a) sec (b) 1.54 sec 7. 10 shows 8. 4 100 600 0, 5 m 9. width = m 10. 4 60 100 0, 1.51 m 11. width = 3 m 1(a) B (b) C (c) D 13(a) 0 roots (b) 1 root (c) roots 14. { 9, 7 }, { 7, 9 } 15. { 6, 4 }, { 4, 6 }