y ax bx c OR 0 then either a = 0 OR b = 0 Steps: 1) if already factored, set each factor in ( ) = 0 and solve
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1 Algebra 1 SOL Review: Quadratics Name 67B Solving Quadratic equations using Zero-Product Property. Quadratic equation: ax bx c 0 OR y ax bx c OR f ( x ) ax bx c Zero-Product Property: if a b 0 then either a = 0 OR b = 0 Steps: 1) if already factored, set each factor in ( ) = 0 and solve ) if NOT already factored, factor the equation completely. Then set each factor in ( ) = 0 and solve 3) put your answers in a set Remember: Solution = Root = X-intercept = Zeros = Answers Solve the following: 1. Find the solution of ( x 3)( x 5) 0. Find the roots of x 10x Find the x-intercepts of y x Find the zeros of f x x x ( ) What is the solution set of 3 x ( x 4) 0 6. What are the zeros of the function f x x x ( ) 3 5
2 7. What is the solution set for the following quadratic equation? x 4x What are the x-intercepts of the graph of the following equation? y x x What are the solutions to 9x x 0? A. B. C., D. 1,3 A. -7 and -1 B. 1 and 7 C. -1 and 7 D. -7 and 1 A. x = -4 or x = -5 B. x = or x = 10 C. x =4 or x = 5 D. x = -5 or x = Which is a zero of the function defined by the following equation? x x f x 11. Which is a zero f the function defined by the following equation? x 6 f x 1. Which is a zero of the function g x 7x 4? A. - B. -1 C. 1 D. A. -6 B. -3 C. D. 3 A. 4 B. 6 C. -6 D Which number is a zero f the function f? f x x x Which quadratic equation has solutions of 5 and 7? 15. Which is a zero of the function f x x 5x 4? A. 0 B. C. 3 D. 6 A x. 5x 0 B x x C x x D x x A. -8 B. -5 C. 3 D. 8
3 16. If f x x, what are the value(s) of x for which f x equals 0? 17. Which of the following is a solution to x x 1 0? 18. What is the solution set for the equation x 4 0? A. - only B. only C. 0 and D. - and 19. What is the solution set for the equation below? x 3x 1 0 A. -1 B. -3 C. - D What are the roots of the quadratic equation shown in the graph? A. 4,1 B., C. 1,4 D. 0,4 1. What are the apparent x- intercepts of the quadratic equation shown in the graph? A., 1 B. C. 1 1, 1,1 D. 1, A., B.,1 C. 1, D. 0,4 A. x = -4 B. x = - C. x = -4 and x = - D. x = -3. Which factors apparently match the parabola in the graph? 3. Which equation apparently matches the parabola in the graph? 4. Plot the zeros of the quadratic equation shown in the graph. A. 1( x 3)( x 1) B. 1(3 x )(x 1) C. 1( x )( x 1) D. 1( x 3)( x 1) A. B. C. D. x 3x 0 x 3x 0 x x 0 3x 4x 4 0
4 Quadratic Standard Form: ax bx c 0 Quadratic Formula: Find a, b, and c then plug into formula and simplify x a b b 4ac 5. Solve the equation by each method listed. Show all steps!! Be precise in your work! y x x 8 6 a) factor/zpp b) quadratic formula c) graphing. vertex = ( ) 6. Solve the equation by each method listed. Show all steps!! Be precise in your work! f x x x ( ) 6 9 a) Factor/ZPP b) quadratic formula c) graphing. vertex = ( )
5 7. Solve the equation by each method listed. Show all steps!! Be precise in your work! x 9 0 a) factor/zpp b) quadratic formula c) graphing. vertex = ( ) 8. Solve the equation by each method listed. Show all steps!! Be precise in your work! x 4 0 a) factor/zpp b) quadratic formula c) graphing. vertex = ( ) Application Word Problems STEPS: READ the problem. DECIDE what it is asking you to find. LIST all knowns and unknowns. SUBSTITUTE in your given values (knowns) into the formula. FACTOR and SOLVE. ANSWER the question.
6 EXAMPLE A (Rectangle/area): Mrs. Newman decides to plant a bulb garden. She wants the length of the rectangular garden to be 3 feet longer than its width. The bulbs that she has will cover 88 square feet of area. Knowing that A l w, what should be the length and width of her garden? Step 1 = draw a rectangle Step = label the width and length Step 3 = write an equation using the formula (use the width and length in your picture) Step 4 = solve by factoring and ZPP Step 5 = answer the question completely Vertical Motion Formula h 16t vt s h = height (when object is on the ground h = 0) t = time in seconds (how long did it take, cannot be zero or negative) v = initial velocity (means your speed.usually in feet/second. There is NO initial velocity if you drop an object) s = initial height (where did you start to throw or drop from) EXAMPLE B (Object thrown): A startled armadillo jumps straight into the air with an initial velocity of 1 ft/sec. Given that the height (in feet) of a projectile can be modeled by the formula h 16t vt s, after how many seconds does the armadillo land on the ground? Step 1 = List all knowns and unknowns : h = t = v = s = Who are you solving for? Step = Substitute the known values into the formula h 16t vt s 16 t t Step 3 = Factor and Solve using ZPP. (Make sure problem = 0 first!) Step 4 = Answer the question.
7 EXAMPLE C (Object dropped): A window washer drops a wet sponge from a height of 64 feet. Given that the height (in feet) of a projectile can be modeled by the formula h 16t vt s, after how many seconds does the sponge land on the ground? (Hint = there is NO initial velocity if you drop an object) Step 1 = List all knowns and unknowns : h = t = v = s = Who are you solving for? Step = Substitute the known values into the formula h 16t vt s 16 t t Step 3 = Factor and Solve using ZPP. (Make sure problem = 0 first!) Step 4 = Answer the question. EXAMPLE D (Complex or multi-solutions): You hit a baseball straight up into the air with an initial velocity of 80 ft/sec when it is 3 feet off the ground. Given that the height (in feet) of a projectile can be modeled by the formula h 16t vt s, after how many seconds does the ball reach a height of 99 feet? Does the ball reach a height of 99 feet more than once? Justify your answer. Step 1 = List all knowns and unknowns : h = t = v = s = Who are you solving for? Step = Substitute the known values into the formula h 16t vt s 16 t t Step 3 = Factor and Solve using ZPP. (Make sure problem = 0 first!) Step 4 = Answer the question.
8 You Try. First decide which of the 4 types of application problem it is. Then solve it. 9. A cat leaps from the ground into the air with an initial velocity of 11 ft/sec. Given that the height (in feet) of a projectile can be modeled by the formula h 16t vt s, after how many seconds does the cat land on the ground? 30. A window washer drops a wet sponge from a height of 16 feet. Given that the height (in feet) of a projectile can be modeled by the formula h 16t vt s, after how many seconds does the sponge land on the ground? 31. A spittlebug jumps into the air with an initial velocity of 10 ft/sec. Given that the height (in feet) of the spittlebug can be modeled by the formula h 16t vt, after how many seconds will the spittlebug land on the ground? 3. If the spittlebug in #31 reaches its maximum height after seconds, how high can it jump?
9 33. Sarah is jumping rope and leaves the ground at an initial velocity of 8 ft/sec. Given that the height (in feet) of a projectile can be modeled by the formula h 16t vt s, after how many seconds does Sarah land on the ground? 34. A cliff diver jumps from a ledge 96 feet above the ocean with an initial velocity of 16 ft/sec. Given that the height (in feet) of a projectile can be modeled by the formula h 16t vt s, how long will it take until the diver enters the water? 35. An athlete throws a discus from an initial height of 6 feet and with an initial velocity of 46 ft/sec. Given that the height (in feet) of a projectile can be modeled by the formula h 16t vt s, after how many seconds does the discus hit the ground?
10 36. You throw a ball into the air with an initial velocity of 31 ft/sec. The ball leaves our hand when it is 6 feet above the ground. You catch the ball when it reaches a height of 4 feet. Given that the height (in feet) of a projectile can be modeled by the formula h 16t vt s, after how many seconds do you catch the ball? 37. The Parthenon in Athens, Greece, is an ancient structure that has a rectangular base. The length of the Parthenon s base is 8 meters more than twice the width. The area of the base (A=lw) is about 170 square meters. Find the length and width of the Parthenon s base. 38. To catch a Frisbee, a dog leaps into the air with an initial velocity of 14 ft/sec. Given that the height (in feet) of a projectile can be modeled by the formula h 16t vt s, after how many seconds does the dog land on the ground?
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