# Solving Quadratics By Factoring (Day 10 1)

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Solving Quadratics by Factoring (Day 10–1)Recall: Factor x 2 2 x 8 completely.What values of x make (x 4)(x 2) 0 true?Graph y x 2 2x 8 .What is the connection?1

Quadratic Equation (Standard Form):Steps for Solving a Quadratic Equation by Factoring Write equation in standard form. Factor the quadratic equation. (GCF, D2PS, X-Box, Grouping) After the problem has been factored we will complete a step called the “T-chart”.Create a T-chart separating the two ( ). Once ( ) are separated, set each ( ) to 0 and solve for the variable. If necessary, check each of the roots in the ORIGINAL quadratic equation.1. Find the roots: f(x) x 2 642. Find the zeroes: 5x 2 35x3. Solve: 2x 2x 2 604. Find the zeroes: 35 x 2 12x5. Find the x-intercepts: g(x) 6x 2 x 22

Classwork 10–11. Solve for the roots: x 2 12x 20 03. Find the zeroes:2. Find the x-intercepts: h(x) 9x 2 1x 2 6x4. Find the zeroes: 10x 2 x 2 55. Solve for the zeroes of y x 2 7x 10 . Then, graph the equation.What is the connection between the zeroesand what you see on the graph?3

6. Tony makes a phone call at a pay phone. The charge is 0.25 for placing the call and 0.10 for each minute. Tony has 2.10 in change in his pocket. Write an inequality thatcan be used to find m, the maximum number of minutes that Tony can talk on the phone.Solve this inequality algebraically to find the maximum number of whole minutes he cantalk on the phone.7. Graph the inequalities and label the solution set with an “S.”3x y 7y y2x 43x4

Solving Quadratics by Completing the Square (Day 10–2)Why is x 2 8x 16 an example of a perfect square trinomial? Hint: Factor it.SOLVING A QUADRATIC EQUATION BYCOMPLETING THE SQUARE1. Rearrange the equation: Get terms with variables on the left hand side.Get c# (constant) by itself on the right hand side.2. If a# 1 then divide through by a#.3. Complete the Square Identify the b#.Take half of b - square it - add it to both sides)(This will form a perfect square trinomial).Write expression as a perfect square trinomial.Simplify the # on the right side of the sign.4. Solve for x. Square root both sidesPut a on the right in front of termSolve for x to find the roots.52x 2 12x 14 0

Solve by completing the square. Express each root in simplest radical form when necessary.1. x 2 8x 62. 3x 2 6x 24 03. 2x 2 8 24x4. If x 2 2 6x is solved by completing the square, an intermediate step would be:(1) (x 3)2 7(3) (x 3)2 11(2) (x 3)2 7(4) (x 6)2 346

Classwork 10–2Solve by completing the square. Express each root in simplest radical form when necessary.1. 3x 2 12x 242. 2x 2 62 4x7

For questions #3 & 4:a) Write a formula for the given sequence.b) Use the formula to find a10 .3. 7, 14, 28, 56, 112, 4. 1, 1.5, 2, 2.5, 3, a)a)b)b)5. Which equation is an example of the use of the associative property of addition?x 7 7 x(3) (x y) 3 x (y 3)(2) 3(x y) 3x 3y(4) 3 (x y) (x y) 3(1)6. Solve and graph the following inequality. Then express your solution in interval notation. 2(x 4) 6x 168

More Completing the Square (Day 10–3)1. Milton made a mistake when beginning to solving 2x 2 8x 4 0 by completing thesquare. Explain and correct the mistake, and then finish solving the equation.2x 2 8x 42 x 2 8 x 16 4 162x 2 8x 16 202. Solve by completing the square and, if necessary, express result in simplest radical form: 4x 2 24x 119

Solve each of the following by completing the square and, if necessary, express result insimplest radical form:3.1 2x 2 x 1224. 4x(x 4) 95. Why is complete the square a difficult method to solve this equation?x 2 3x 810

Classwork 10–3Solve each of the following by completing the square. Express your answer in simplestradical form when necessary.2. x 2 4 x 121. 3x 2 18x 57 03. Solve for V in terms of P and R:P V2R11

4. Which property is illustrated in the following equation?(x 6)(8 x) 8(x 6) x(x 6)(1) Distributive property(2) Associative property of addition(3) Associative property of multiplication(4) Commutative property of multiplication5. Which equation represents the line that passes through the points (–1, –2) and (3, 10)?(1) y 3 x 1(3) y 4 x 2(2) y 3 x 1(4) y 4 x 26. Write a recursive and explicit equation for the sequence 7, 21, 63, 189, 7. An online music club has a one-time registration fee of 13.95 and charges 0.49 to buyeach song. If Emma has 50.00 to join the club and buy songs, what is the maximumnumber of songs she can buy?12

The Quadratic Formula (Day 10–4)Completing the square and factoring are not always the best method to use when solving aquadratic equation.Why are completing the square and factoring not good options for the quadratic below?7p2 12p 4 0Steps for using the Quadratic Formula Get equation equal to ZERO!!! Put equation in standard form: Identify the a, b, and c #’s. Plug into the formula and simplify.To remember formula sing/hum the phase below to the “pop goes the weasel song”“x ’s negative b, plus or minus the square root of b2 minus 4 a c, all over 2 a”******WRITE THE FORMULA DOWN AS YOU SING THE SONG******Quadratic Formula:13

Solve for the roots/zeroes of each of the following quadratic equations using the quadraticformula. If necessary, express your answers in simplest radical form.1. 2x 2 18 9x2. h(x) x 2 5x 33. x 2 12x 204. 2p2 4p 114

Classwork 10–4Solve for the roots/zeroes of each of the following quadratic equations using the quadraticformula. If necessary, express your answers in simplest radical form.2. k(x) 2x 2 8x 71. x 2 2x 123. The perimeter of a triangle can be represented by the expression 5x 2 10x 8 . Write apolynomial that represents the measure of the third side.15

4. Paula just bought a new car for 18,600. She looked up on the internet that her model isexpected to depreciate in value by 18.5% every year. If she plans on owning the car for 3years, what should she expect the value to be after the 3 years?5. Jack Eichel and Ryan O’Reilly are hungry for pizza and breadsticks. They order food fromthe same restaurant at the same prices. Jack orders three large pizzas and two orders ofbreadsticks and pays 44. Ryan orders five large pizzas and four orders of breadsticks for 76. How much do a large pizza and an order of breadsticks cost?6. Given the equation (x 7)(x 2) 0 , what is the smaller root?16

More with the Quadratic Formula (Day 10–5)Quadratic Formula:Solve for the roots/zeroes of the following quadratic equations using the quadratic formula. Ifnecessary, get your answer in simplest radical form.2. f(x) 5x 2 3 x 21. 3x 2 8x 1217

Solve each of the following using the most appropriate method.4. Find the x-intercepts: f(x) 2 x 2 x3. Find the roots: 5x 2 125 05. Which method is best to solve the equation x 2 7x 5 ? Why?(1) Complete the square since the “b” value is positive.(2) Complete the square since the “b” value is even.(3) X-box factoring since it can factor.(4) Quadratic formula – other techniques failed or are very difficult.18

Classwork 10–5Solve for the roots/zeroes of the following quadratic equations using the quadratic formula. Ifnecessary, get your answer in simplest radical form.2. 2b2 8 4b1. x 2 3x 8 03. Matt made a mistake when solving 2x 2 5x 2 0 by the quadratic formula. Explain andcorrect the mistake.x ( 5) ( 5)2 4( 2 )( 2 )x 5 25 16x 5 9x 5 3x 2, 8 19

4. Simplify the following and expressing all answers with only positive exponents.b) 5y(2y 4 7y 3 ) 4y 3 (y 2 2y)a) ( 3x 5 y 5 )(9xy 2 z 4 )3ab 2 4a2 bd)ab2c) (3 x 6)5. Solve the system of equations graphically.y2y 3 x 8 x y 1x20

Mixed Quadratic Problems (Day 10–6)METHODREASON TO USE THIS METHOD21QUICK PROCEDURE

1. A rocket carrying fireworks is launched from a hill 80 feet above a lake. The rocket will fallinto the lake after exploding at its maximum height. The rocket’s height above thesurface of the lake is given by h(t) 16t 2 64t 80 , where t is the time in seconds and h isthe height in feet.a) What does t represent?b) What does h(t) represent?c) What is the input of the function?d) What is the output of the function?e) How long will it take for the rocket to reach 128 feet?f)Find the height at 0 seconds. What is the graphic name of this point?g) Find the height at 2 seconds.22

2. The area of a rectangular playground enclosure at Happy Times Nursery School is 600 sq.meters. The length is 25 meters longer than the width. Find the dimensions of theplayground.3. Collin is building a deck on the back of his house. He has enough lumber for the deck tobe 144 square feet. The length should be 10 feet more than its width. What should thedimensions of the deck be?23

Classwork 10–61. After t seconds, a ball tossed in the air from the ground level reaches a height of h feetgiven by the equation h(t) 16t 2 144t .a) What is the height of the ball at 4 seconds?b) After thrown, when will the ball hit the ground?2. The length and width of a rectangle are consecutive odd integers. If the area of therectangle is 63 in2, find the dimensions of the rectangle. Only an algebraic solution will beaccepted.24

3. Consider the quadratic equation 3x 2 5 10x .a) State two possible methods to solve for the roots.b) Using one of the ways stated above, solve the quadratic equation 3x 2 5 10x insimplest radical form.4. Simplify:b) 3 x 2 125x 6a) 2 805. Factor:b) 14x 3 y 2 10x 2 y 5a) 4 x 2 8125

Exam Quadratic Problems (Day 10–7)1. The Demon Drop at Cedar Point in Ohio takes riders to the top of a tower and drops them60 feet. A function that approximates this ride is h 16t 2 64t 60 , where h is the heightof the riders in feet and t is the time in seconds. To the nearest tenth, how many secondsdoes it take for riders to hit the ground?2. Consider the quadratic equation w(x) (2 x)(2x 1) 3x 2 48 .a) Simplify w(x) and write it as a trinomial.b) Solve for x when w(x) 0 .26

3. Tammy found the zeroes of the function f(x) to be –3 and 8. Write a quadratic equationthat could represent Tammy’s function f(x) .4. Milton made a mistake when solving x 2 6x 3 0 by the quadratic formula. Circle,explain and correct the mistake. ( 6) ( 6)2 4(1)( 3 )x 2(1)x 6 36 122x 6 242x 6 2 62x 3 2 627

Classwork 10–71. Emma made a mistake when solving x 2 2x 8 0 by completing the square. Explainand correct the mistake.x 2 2x 8(x 2 2 x 1 ) 8 1( x 1)2 9x 1 3x 22. Solve the equation for y:(y 3)2 4y 123. The roots of a function are x 2 and x 9 . Write a possible quadratic function for theseroots.28

4. The senior class at Bay High School buys jerseys to wear to the football games. The costof the jerseys can be modeled by the equation C(x) 0.1x 2 2.4x 25 , where C(x) is theamount it costs to buy x jerseys. How many jerseys can they purchase for 500?5. Using the given graph, state the inequality that hasthe negative slope.6. A town’s population is currently at 100,000 and is growing at an annual rate of 5%. Whatwill the population be in 8 years?29

More Exam Quadratic Problems (Day 10–8)1. Find the zeroes of f(x) (x 3)2 49 algebraically.2. The height, H, in feet of an object dropped from the top of a building after t seconds isgiven by H(t) 16t 2 144 .a) Determine the height of the building.b) How many feet did the object fall between one and two seconds after it wasdropped?c) Determine, algebraically, how many seconds it will take for the object to reach theground.30

3. Amy solved the quadratic equation 2x 2 5x 42 0 . She stated that the solutions to the7equations wereand –6. Do you agree with Amy’s solutions? Explain why or why not.24. The length and width of a rectangle are consecutive even integers. If the area of therectangle is 224 in2, find the dimensions of the rectangle. Only an algebraic solution willbe accepted.31

Classwork 10–81. The solution of the equation (x 3)2 7 is:(1) 3 7(3) 7 3(2) 3 7(4) 7 32. Keith determines the zeroes of the function f(x) to be 6 and –5. Which of the followingcould be Keith’s function?(1) f(x) (x 5)(x 6)(3) f(x) (x 5)(x 6)(2) f(x) (x 5)(x 6)(4) f(x) (x 5)(x 6)3. When solving the equation x 2 8x 7 0 by completing the square, which equation is astep in the process?(1) (x 4)2 9(3) (x 4)2 23(2) (x 8)2 9(4) (x 8)2 234. Rhiannon was asked to solve this word problem: “The product of two consecutive evenintegers is 224. What are the integers?” What type of equation should she create to solvethis problem?(1) linear(3) quadratic(2) exponential(4) absolute value32

5. Using the diagram below, write a formula to represent the number of blocks in the nthdiagram.6.Tabitha solved x 2 2x 8 0 by the quadratic formula below. Is her solution correct orincorrect? Explain your reasoning.x (2) (2)2 4(1)( 8 )2(1)x 2 4 322x 2 362x 2 62x 2, 4 7.Solve for x:x51 461233

1 Solving Quadratics by Factoring (Day 10 . If necessary, express your answers in simplest radical form. 1. x2 2x 12 2. k(x) 2x2 8x 7 3. The perimeter of a triangle can be represented by the expression 5x2 10x 8. Write a polynomial that represents the measure of the third side. 16

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