Lesson 3.4 Exercises, pages

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1 Lesson 3. Exercises, pages 17 A. Identify the values of a, b, and c to make each quadratic equation match the general form ax + bx + c = 0. a) x + 9x - = 0 b) x - 11x = 0 Compare each equation to ax bx c 0 a 1, b 9, c a, b 11, c 0 c) 11x - 3x + = 0 d) 3.x +.1 = 0 3x 11x 0 a 3, b 11, c a 3., b 0, c.1 P DO NOT COPY. 3. Developing and Applying the Quadratic Formula Solutions 3

2 5. Simplify each radical expression. ; 1 a) _ b) 5; or 1 - ; 0 _ 5 ( _ 5) _ 5 3 ; 5 c) 3_3 5 d) 3(1_ 5) 1_ 5 1 ; 1_ 7 (_ 7) _ 7 B. Solve each quadratic equation. a) x + x + = 0 b) x - 10x + 17 = 0 For each equation, substitute for a, b, and c in: x b_ b ac a 1, b, c a 1, b 10, c 17 x ( )_ ( ) (1)() (1) x _ 0 x _ 5 x 3_ 5 x ( 10)_ ( 10) (1)(17) (1) x 10_ 3 x 10_ x 5_ c) x + x - 3 = 0 d) x - x - 1 = 0 a 1, b, c 3 x _ (1)( 3) (1) x _ x _ 7 x _ 7 a, b, c 1 x ( )_ ( ) ()( 1) () x _ 1 x _ 3 x 1_ 3 3. Developing and Applying the Quadratic Formula Solutions DO NOT COPY. P

3 7. Solve each quadratic equation. a) 3x = x + 1 b) x - 1 = -7x For each equation, substitute for a, b, and c in: x b_ b ac 3x x 1 0 a 3, b, c 1 x _ ( ) (3)( 1) (3) x _ x _ 7 x _ 7 3 x 7x 1 0 a, b 7, c 1 x 7_ 7 ()( 1) () x 7_ 5 c) x(x - 3) = (x - 3) + 1 d) (x + 1) + = 0 x x x x 10x 11 0 a, b 10, c 11 x 10_ ( 10) ()(11) () x 10_ 1 x 10_ 3 x 5_ 3 x x 1 0 x x 3 0 a, b, c 3 x _ ()(3) () x _ 3 The radicand is negative, so there are no real roots.. A student wrote the solution below to solve this quadratic equation: x - 3 = 7x x = b - ac -b ; (-7) - ()(-3) x = -(-7) ; 73 x = 7 ; () Identify the error, then write the correct solution. The student wrote an incorrect quadratic formula. The correct solution is: x 7_ ( 7) ()( 3) () x 7_ 73 P DO NOT COPY. 3. Developing and Applying the Quadratic Formula Solutions 5

4 9. a) Solve each equation by factoring. i) 3x = 11x + 0 ii) 1x + x = 15 3x 11x 0 0 (3x )(x 5) 0 x or x 5 3 1x x 15 0 (x 3)(x 5) 0 x 3 or x 5 b) Solve each equation in part a using the quadratic formula. For each equation, substitute for a, b, and c in: x b_ b ac i) 3x 11x 0 0 ii) 1x x 15 0 a 3, b 11, c 0 a 1, b, c 15 x 11_ ( 11) (3)( 0) (3) x _ (1)( 15) (1) x _ 7 x 11_ 31 x 11_19 x _ x 5 x Or, x Or, x 3 3 c) Which method do you prefer and why? I prefer to factor when the numbers are small because it is quicker. I prefer to use the quadratic formula when the numbers are large and I have many factors to guess and test. 10. For each equation, choose a solution strategy, justify your choice, then solve the equation. a) x + 9x + = 0 b) x + 7x - 30 = 0 I use the formula because I cannot factor. a, b 9, c in: x b_ b ac x 9_ 9 ()() () x 9_ 17 I can factor. (x 3)(x 10) 0 x 3 or x Developing and Applying the Quadratic Formula Solutions DO NOT COPY. P

5 c) (x + ) = 1 d) + 5.x - 1.x = 0 x 1x 0 I use the formula because I cannot factor. a 1, b 1, c in: x b_ b ac x 1_ 1 (1)() (1) x 1_ x 1_ 3 x _ 3 I use the formula because I cannot factor. a 1., b 5., c in: x b_ b ac x 5._ 5. ( 1.)() ( 1.) x 5._ 9.7. x 5._ Solve each quadratic equation. Give the solution to 3 decimal places. 1 a) b) -x + 3 x - 3 x - 3x + 1 = 0 5 = 0 Multiply by 1. Multiply by 10. x 3x 3 0 0x 15x 0 a, b 3, c 3 a 0, b 15, c in: x b_ b ac in: x b_ b ac x 15_ 15 ( 0)( ) x 3_ ( 3) ()(3) () ( 0) x 15_ 15 x 3_ 1 0 There are no real roots. x 3 1 so, x.91 Or, x 3 1 so, x 0.0 c).9x + 1x - 0. = 0 d).1x = 1.x + 3 a.9, b 1, c 0. in: x b_ b ac x 1_ 1 (.9)( 0.) (.9) x 1_ 159. x so, x 0.05 Or, x so, x.51 a.1, b 1., c 3 in: x b_ b ac x 1._ ( 1.) (.1)( 3) (.1) x 1._.. x 1... so, x Or, x 1... so, x 0.93 P DO NOT COPY. 3. Developing and Applying the Quadratic Formula Solutions 7

6 1. Solve each radical equation. Check for extraneous roots. a) + 5x = 3x b) x = x x 3x ( 5x) (3x ) 5x 9x 1x 9x 17x 0 a 9, b 17, c in: x b_ b ac x 17_ ( 17) (9)() (9) x 17_ 15 1 Use a calculator to check: The root is: x x 3 x 10 (x 3) ( x 10) x 1x 9 x 10 x 10x 1 0 a, b 10, c 1 in: x b_ b ac x 10_ 10 ()( 1) () x 10_ 11 x 10_ 9 x 5_ 9 Use a calculator to check: The root is: x a) Solve this equation using each strategy below: x - 10x - i) the quadratic formula ii) completing the square = 0 x 10x 0 a 1, b 10, c in: x b_ b ac x 10_ ( 10) (1)( ) (1) x 10_ 19 x 10_ x Or, x x 10x 0 x 10x x 10x 5 5 (x 5) 9 x 5 9 x 5_7 x 1 or x iii) factoring x 10x 0 (x 1)(x ) 0 x 1 or x b) Which strategy do you prefer? Is it the most efficient? Explain. Sample response: I prefer factoring; it is the most efficient because it takes less time and less space. 3. Developing and Applying the Quadratic Formula Solutions DO NOT COPY. P

7 1. A person is standing on a bridge over a river. She throws a pebble upward. The height of the pebble above the river, h metres, is given by the formula h = + 9t -.9t, where t is the time in seconds after the pebble is thrown. a) When will the pebble be 0 m above the river? Give the answer to the nearest tenth of a second. In h 9t.9t, substitute h 0, then solve for t. 0 9t.9t 0 9t.9t a.9, b 9, c in: t b_ b ac t 9_ 9 (.9)() (.9) t 9_ 19. Ignore the negative root since t cannot be negative. t t The pebble is 0 m above the river after approximately. s. b) When will the pebble be 30 m above the river? Give the answer to the nearest tenth of a second. In h 9t.9t, substitute h 30, then solve for t. 30 9t.9t 0 9t.9t a.9, b 9, c in: t 9_ 9 (.9)( ) (.9) t 9_. t 9., or t b_ b ac t 9., or The pebble is 30 m above the river after approximately 0. s and 1.1 s. c) Why are there two answers for part b, but only one answer for part a? There are two answers for part b because the stone is 30 m above the river on its way up and on its way down. There is only one answer for part a because the stone is only 0 m above the river on its way down. P DO NOT COPY. 3. Developing and Applying the Quadratic Formula Solutions 9

8 15. A car was travelling at a constant speed of 19 m/s, then accelerated for 10 s. The distance travelled during this time, d metres, is given by the formula d = 19t + 0.7t, where t is the time in seconds since the acceleration began. How long did it take the car to travel 00 m? Give the answer to the nearest tenth of a second. In d 19t 0.7t, substitute d 00, then solve for t t 0.7t t 0.7t a 0.7, b 19, c 00 in: t 19_ 19 (0.7)( 00) (0.7) t 19_ Ignore the negative root since t cannot be negative. t t The car travelled 00 m in approximately.1 s. t b_ b ac 1. Josie s rectangular garden measures 9 m by 13 m. She wants to double the area of her garden by adding equal lengths to both dimensions. Determine this length to the nearest centimetre. Let the length added be x metres. The new width, in metres, is: x 9 The new length, in metres, is: x 13 The new area, in square metres is: (x 9)(x 13) The original area is: (9)(13), or 117 m The new area is: (117 m ) 3 m An equation is: (x 9)(x 13) 3 x x x x a 1, b, c 117 in: x _ (1)( 117) (1) x b_ b ac x _ 95 Ignore the negative root since x cannot be negative. x 95 x.7... The length added is approximately.3 m Developing and Applying the Quadratic Formula Solutions DO NOT COPY. P

9 C a) Solve this equation x - 3 x - 1 = 0 in the two ways described below: i) Substitute the given coefficients and constant in the quadratic formula. 1 x 3 x 1 0 a 1 in:, b 3, c 1 x b_ b ac x 3 _ B a 3 b a 1 b( 1) x 3 _ 1 1 x 3 _ 1 1 b ii) Multiply the equation by a common denominator to remove the fractions, then substitute in the quadratic formula. 1 x 3 x 1 0 x 3x 0 Multiply by. a, b 3, c in: x 3_ ( 3) ()( ) () x 3_ 1 x b_ b ac b) Which strategy in part a do you prefer? Explain why. I prefer the strategy in part ii because it is easier to work with integers than fractions. 1. This quadratic equation has only one root: x + x + d = 0 Use the quadratic formula to determine the value of d. Explain your strategy. x x d 0 a, b, c d in: x b_ b ac x _ ()(d) () The equation has only one root, so the radicand must be 0. 3 d 0 d.5 P DO NOT COPY. 3. Developing and Applying the Quadratic Formula Solutions 31

10 19. a) Solve this quadratic equation by expanding, simplifying, then applying the quadratic formula: (x - 5) - 7(x - 5) - = 0 x 0x 50 7x 35 0 x 7x 3 0 a, b 7, c 3 in: x 7_ ( 7) ()(3) () x 7_ 5 x b_ b ac b) Solve the equation in part a using the quadratic formula without expanding. (x 5) 7(x 5) 0 a, b 7, c in: x 5 7_ ( 7) ()( ) x 5 7_ 5 x 5 7_ 5 x 7_ 5 x 5 b_ b ac 0. a) Is this equation quadratic: x + x = 1? Justify your response. The equation is not quadratic because it contains an x -term. b) Describe a strategy you could use to solve the equation in part a. Write the equation as: (x ) (x ) 1 0, then use the quadratic formula. c) Solve the equation in part a. a 1, b 1, c 1in: x 1_ 1 (1)( 1) (1) x 1_ 5 x 1 5 x b_ b ac Since x is positive, ignore the negative root Developing and Applying the Quadratic Formula Solutions DO NOT COPY. P