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Lesson 7.1: Solving Quadratic Equations by Graphing, Factoring, and Square Root MethodLearning Goals:How do we solve a quadratic equation by graphing?How do we solve a quadratic equation by factoring?How do we solve a quadratic equation by the square root method?Warm up: 1) Solve the given equation 3 different ways.x2-4=0GraphicallyFactoringSquare Root MethodDOPSx-2x+2=0x-2=0 x+2=0x=±2x2=4x2=4x=±22) Express in simplest radical form:-25-1*255i-32-16*24i2452*55*5255Advice to solving Quadratic Equations:Express quadratic in standard form ax2+bx+c=0Determine appropriate methods to solve quadratic equations.1. Solve the following equation by factoring:x2-48=2xSet =0x2-2x-48=0x-8x+6=0x-8=0 x+6=0x=8 &-62. Solve for x and express in simplest form:3x2+9=03x2=-9x2=-3x=±i33486150322580003. Use graphing to find the roots of x2-2x=81. Solve for a: a2=5aa2-5a=0 set =0aa-5=0 GCF =aa=0 a-5=0 a=0 & 5 2. Find the zeros of fx=5x2-8x-40=5x2-8x-4AC method0=5x2-10x+2x-45-4=-200=5xx-2+2(x-2) -10, 20=5x+2x-2 x=-25 & 2 3. Solve for x: (x+3)2=16x+32=16 x+3=±4 x=-3±4 -3+4 -3-4 x=1 & -7 4. Algebraically solve for x in simplest form: 3x2+4=03x2=-4 x2=-43 x=±-43=±2i3 *33 or ±2i33 5. Find the roots of the equation: 2x2+x-3=(x-1)(x+2)2x2+x-3=x2+2x-x-2 2x2+x-3=x2+x-2 x2-1=0 x=±1 6. Find the zeros of x-4=5x Cross Multiply!xx-4=5 x2-4x=5 x2-4x-5=0 x-5x+1=0 x=5 & -1 7. Solve for x: 14(x-6)2=8x-62=32 x-6=±32 x=6±42 8. Solve for x: (x+1)2-87=x2x+1x+1-87=x2 x2+2x+1-87=x2 2x-86=0 2x=86 x=43 Check for understanding…1. On a test, Samantha says that the roots of the equation x2+2x-3=0 are -1 and 3. Before she hands in her test, she wants to check over her work. Show the check that Samantha could do on her test and state whether or not she is correct. If she is not correct, give the correct solution.(-1)2+2-1-3=0 (3)2+23-3=01-2-3=0 9+6-3=0-4≠0 12≠0Therefore she is not correct!x+3x-1=0 x=-3 & 1 2. Phil’s teacher gave the class the quadratic function fx=(x-2)2-4.a) State two different methods Phil could use to solve the equation fx=0.b) Using one of the methods stated in part a, solve fx=0 for x.Factoring, Quadratic Formulafx=x-2x-2-4 fx=x2-4x+4-4 fx=x(x-4) x=0 & 4 Homework Lesson 7.1: Solving Quadratic Equations by Graphing, Factoring, and Square Root Method1. Solve for x: x-4=-3x2. Find the roots of 3x2-5x=36-2x3. Solve for x in simplest form: 3x2+25=04. Solve for x: 5(x-1)2=5033337500005. Use graphing to solve the given equation for x: x2+5x+6=06. Find the zeros of fx=4x2-16-207. Solve for x in simplest form: 2x+3x-4=2x2+13x+158. Solve for x: x2+x-1=(-4x+3)2Lesson 7.2: Solving Quadratics by Completing the SquareLearning Goal: How do we solve a quadratic equation by completing the square?Practice: Find the value of c that makes the expression a perfect square trinomial. Then write the expression as the square of a binomial.x2+14x+cc=1422=72=49x2+14x+49x+7x+7(x+7)2x2-9x+cc=-922=814x2-9x+814x-922Steps to Completing the SquareThe "a" coefficient must equal 1 (divide all terms by "a").Isolate the x-terms (move "c" to the other side).Take b22 and add that number to each side.Factor the perfect square trinomial and express it as (x+a)2.Solve for x by using the square root pleting the Square when a=1Example 1: Solve x2-10x+13=0 by completing the square.SolutionStepsx2-10x+13=0Write original equation.x2-10x=-13Write left side in the form x2+bx, I say “isolate the x”x2-10x+c=-13+cc=-1022=-52=25Complete the square, I say “divide by two and square it”x2-10x+25=-13+25(x-5)2=12Write left side as a binomial squared.(x-5)2=12Take the square roots of each side.x-5=±12x=5±12Solve for x.x=5±23SimplifyPractice: Solve xx+3=-2 by completing the square, and express the results in simplest form. x2+3x=-2 x2+3x+c=-2+c c=322=94 x2+3x+94=-2+94 x+322=14 x+32=±12 x=-32±12 x=-1 & -2 Completing the Square when a≠1Example 1: Solve 3x2-12x+27=0 by completing the square.SolutionSteps3x2-12x+27=0Write original equation.x2-4x+9=0Get a=1before you can complete the square!Divide each side by the coefficient of x2.x2-4x=-9Write left side in the form x2+bx.x2-4x+c=-9+cc=-422=-22=4Complete the square.x2-4x+4=-9+4(x-2)2=-5Write left side as a binomial squared.(x-2)2=-5Take the square roots of each side.x-2=±-5x=2±-5Solve for x.x=2±i5SimplifyPractice: Solve 4x2-20x=-9 by completing the square, and express the results in simplest form. x2-5x=-94 Isolate the x and make sure a=1x2-5x+c=-94+c c=-522=254 x2-5x+254=-94+254 x-522=4 x-52=±2 x=52±2 x=92 & 12 Practice: Solve the following equations by completing the square, and express the results in simplest form.a) x2-9x-1=0b) 5xx+6=-50x2-9x=1 5x2+30x=-50x2-9x+c=1+c x2+6x=-10c=-942=8116 x2+6x+c=-10+cx2-9x+8116=1+8116 c=622=(3)2=9x-942=9716 x2+6x+9=-10+9x-94=±974 x+32=-1x=94±974 x+3=±ix=-3±iAnalysis Question: Which equation has the same solutions as x2+6x-7=0?1. (x+3)2=2 2. (x-3)2=2 3. (x-3)2=16 4. (x+3)2=16x2+6x=7 x2+6x+c=7+c c=622=(3)2=9 x2+6x+9=7+9 (x+3)2=16 Homework 7.2: Solving Quadratics by Completing the Square1. Find the value of c that makes the expression a perfect square trinomial. Then write the expression as the square of a binomial.x2+10x+cy2-7y+c2. Solve the following equations by completing the square and express the results in simplest form.a. x2+6x+3=0b. tt-8=5c. 7y2+28y+56=0d. 4x2-30x=-9-10x3. Determine whether you would use factoring or the square root method to solve each equation. Then solve the equation.a. x2-4x-21=0b. (x+4)2=16c. 5x2-10x+1=2x2-5x+13d. (x+3)2=17x+214. Error Analysis: Describe and correct the error in solving the equation below4x2+24x-11=04x2+6x=114x2+6x+9=11+94(x+3)2=20x+32=5x+3=±5x=-3±55. Determine the roots of the following equations graphically:a. x2-5x-14=0b. x2=9 6. The zeros of the function fx=(x+2)2-25 are1) -2 and 5 2) -3 and 7 3) -5 and 2 4) -7 and 37. A student was given the equation x2+6x-13=0 to solve by completing the square. The first step that was written is shown below.x2+6x=13The next step in the student’s process was x2+6x+c=13+c. State the value of c that creates a perfect square trinomial. Explain how the value of c is determined.Lesson 7.3: Solving Quadratics by Using the Quadratic FormulaLearning Goals:What is the quadratic formula?How do we solve a quadratic equation by using the quadratic formula?Do Now: Solve for x by completing the square and express your answer in simplest radical form:2x2+2=6x2x2-6x+2=0 x2-3x+1=0 Get a=1x2-3x=-1 x2-3x+c=-1+c c=-322=94 x2-3x+94=-1+94 x-322=54 x-32=±52 x=32±52 The Quadratic FormulaPreviously, you solved quadratic equations by completing the square.In this lesson, you will learn about solving quadratic equations by using a formula that is derived by completing the square for the general equation ax2+bx+c=0.The formula is called the Quadratic Formula. x=-b±b2-4ac2a On Reference SheetExample: Solve for x by using the quadratic formula and express your answer in simplest radical form:2x2+2=6x2x2-6x+2=0 Set =0a=2, b=-6, c=2 x=-b±b2-4ac2a=-(-6)±(-6)2-4(2)(2)2(2) x=6±36-164=6±204 x=6±254=3±52 9525156210Summary of Quadratic FormulaWhat is the quadratic formula? x=-b±b2-4ac2aWhen should you use the quadratic formula? When you cannot solve by factoring0Summary of Quadratic FormulaWhat is the quadratic formula? x=-b±b2-4ac2aWhen should you use the quadratic formula? When you cannot solve by factoringSteps to Using the Quadratic FormulaSet quadratic equation equal to zero (ax2+bx+c=0)Identify the a, b, and c coefficientsSubstitute a, b, and c into the quadratic formulaSimplify the formula carefullyLook to simplify and reduce (always start with radical first)Model Problem: Solve for q: 2q2-8=3q2q2-3q-8=0 a=2, b=-3, and c=-8 q=-(-3)±(-3)2-4(2)(-8)2(2)=3±9+644 q=3±734 Directions: Use the quadratic formula to solve each equation.1. Solve for m: 13m2+2m+8=513m2+2m+3=0 313m2+2m+3=0 Easier to eliminate the fraction!m2+6m+9=0 a=1, b=6, and c=9 m=-(6)±(6)2-4(1)(9)2(1)=-6±36-362 m=-3 2. Solve for x: 5x-7x2=3x+4-7x2+2x-4=0 -1(-7x2+2x-4=0) Easier to have a positive coefficient for a!7x2-2x+4=0 a=7, b=-2, and c=4 x=-(-2)±(-2)2-4(7)(4)2(7)=2±-10814 x=2±6i314=1±3i37 Application of the Quadratic FormulaBarb pulled the plug in her bathtub and it started to drain. The amount of water in the bathtub as it drains is represented by the equation L=-5t2-8t+120, where L represents the number of liters of water in the bathtub and t represents the amount of time, in minutes, since the plug was pulled.How many liters of water were in the bathtub when Barb pulled the plug? Show your reasoning. L=?, t=0 because no time has passed!L=-502-80+120 L=120 Determine, to the nearest tenth of a minute, the amount of time it takes for all the water in the bathtub to drain.t=?, L=0 no more water!0=-5t2-8t+120 -10=-5t2-8t+120 0=5t2+8t-120 a=5, b=8, and c=-120 t=-(8)±(8)2-4(5)(-120)2(5)=-8±64+240010=-8±246410 t=4.2 & -5.8 (but omit because time can't be negative) Homework 7.3: Solving Quadratics by Using the Quadratic Formula1. Solve the following equations by using the quadratic formula and express the results in simplest form.a. x2+2x-8=0b. 2y2+3y-5=42. Matt’s rectangular patio measures 9 feet by 12 feet. He wants to increase the patio’s dimensions so its area will be twice the area it is now. He plans to increase both the length and the width by the same amount, x. Find x, to the nearest hundredth of a foot.3. Determine the best method to solve each equation and use it to find all values of x in simplest form.a. x2-2x=12b. (x-2)2=8c. 2x2-54=12xd. 5x2+38=34. Find all real solutions to the equation x2-6x+32x2-4x-7=0Lesson 7.4: Solving Systems of Equations GraphicallyLearning Goals:How can we change an equation of a circle from standard form to center-radius form?How can we solve a quadratic/linear system of equations graphically?Warm-Up:1. Suppose you were given an equation for a circle and an equation for a line. What possibilities are there for the two figures to intersect? Sketch a graph for each possibility.2. Suppose you were given an equation for a parabola and an equation for a line. What possibilities are there for the two figures to intersect? Sketch a graph for each possibility.3. State the center and radius of the given circles:a) (x-1)2+(y-2)2=25Center: 1, 2 & r=25=5 b) (x+5)2+(y-2)2=29 Center: -5, 2 & r=29c) x2+(y-2)2=4Center: 0, 2 & r= 4=24. Rewrite x2+y2-4x+2y=-1 by completing the square in both x and y. Describe the circle represented by this equation.x2-4x+y2+2y=-1-422=(-2)2=4 222=1x2-4x+4+y2+2y+1=-1+4+1(x-2)2+(y+1)2=4Center: (2, -1) and Radius=25. The equation of a circle is x2+y2+6y=7. What are the coordinates of the center and the length of the radius of the circle?x2+y2+6y=7 622=(3)2=9x2+y2+6y+9=7+9x2+(y+3)2=16Center: (0, -3) and Radius=46. If the equation of a circle is 2x2+2y2-32x+12y+52=0, find the length of the radius and the coordinates of the center of the circle.2x2+2y2-32x+12y+52=02x2+y2-16x+6y+26=0x2-16x+y2+6y=-26-1622=(-8)2=64622=(3)2=9x2-16x+64+y2+6y+9=-26+64+9x-82+y+32=47Center: (8, -3) and Radius=477. If the equation of a circle is 4x2+4y2-32x+8y+16=0, find the length of the radius and the coordinates of the center of the circle.4x2+4y2-32x+8y+16=04x2+y2-8x+2y+4=0x2-8x+y2+2y=-4-822=(-4)2=16222=(1)2=1x2-8x+16+y2+2y+1=-4+16+1x-42+y+12=13Center: (-4, 1) and Radius=13Remember to put the 4 back in: 4x-42+4y+12=52Graphing Systems of Equations:1. Graph the line given by 3x+4y=25 and the circle given by x2+y2=25. Find all solutions to the system of equations.261937524765003x+4y=254y=-3x+25 y=-34x+254 y=-34x+6.25 x2+y2=25 Center: 0, 0 r=5 Solution: (4, 3)2905125222885002. Graph the line given by x+y=-2 and the quadratic curve given by y=x2-4. Find all solutions to the system of equations.Solutions: -2, 0 & (1, -3)2895600114300003. Find all solutions to the following system of equations. 5y-5x=30 x2+y2+4x-2y-4=0 y=x+6 x2+4x+y2-2y=4 (x+2)2+(y-1)2=9 Center: -2, 1 r=3 Solutions: -5, 1 & (-2, 4)2990850375920004. Find all values of k so that the following system has two solutions.x2+y2=25y=kCenter: 0, 0 r=5 -5<k<5 Real-World Application:29718003111500An asteroid is moving in a parabolic arc that is modeled by the function px=x2-4x+9 where x represents time. A laser is on the path of fx=2x+4. When will the laser first hit the asteroid?(1) 0, 9 & (1, 6)(2) 1, 6 & (5, 14)(3) 1, 6(4) 2, 5Homework 7.4: Solving Systems of Equations Graphically1. Which of the following systems of equations has exactly one point of intersection?(1) y=x2-5x+7 and y-1=2x(2) y-4x2=x-3 and y=3(3) y-x2=2x+4 and y=3(4) y+9x2=-8 and y=-12. Given the equation of a circle is 3x2+3y2-12x+30y-10=0, state the length of the radius and coordinates of the center.3438525312420003. Find all values of k so that the following system has exactly one solution. Illustrate with a graph.y=5-x-32y=k33432750004. Find all solutions to the following system of equations.y+2x=3y=x2-6x+3303847552705005. Solve the following system of equations graphically.2x+y=15x-22+y-12=25Lesson 7.5: Solving Systems of Equations AlgebraicallyLearning Goals:How can we solve a quadratic/linear system of equations algebraically?How is the solution to a quadratic/linear system related to its graphical solution?Warm-Up: Solve the system x2+y2=9 and x-3y=3 graphically.30480005715000Center: 0, 0 & r=3 -3y=-x+3 y=13x-1 Solutions: 3, 0& ?What is difficult about solving this graphically? Hard to draw a circle without a compass-95250321945Steps to Solving a System of Equations AlgebraicallyGet one equation to be written as y= or x=Substitute this equation into the other equation for x or ySimplify this equation and set it equal to zero.Use one of the methods for solving a quadratic to solve for the variableFactoring, square root method, completing the square, quadratic formulaPlug your answer back into one of the equations to solve for the other variable.Steps to Solving a System of Equations AlgebraicallyGet one equation to be written as y= or x=Substitute this equation into the other equation for x or ySimplify this equation and set it equal to zero.Use one of the methods for solving a quadratic to solve for the variableFactoring, square root method, completing the square, quadratic formulaPlug your answer back into one of the equations to solve for the other variable.1. Find all solutions of the system of equations algebraically:x2+y2=9x-3y=3Substitute: x=3y+3(3y+3)2+y2=9 Now find x-values!3y+33y+3+y2=9 x-30=3x-3-95=39y2+18y+9+y2=9 x-0=3x+275=310y2+18y=0 x=3x=-2.4 or -1252y5y+9=0 y=0 and y=-95 Solutions: 3, 0 and -125,-95How is the solution to a system or equations related to its graphical solution?2. Find all solutions of the system of equations algebraically:y2-2x2=6y=-2xSubstitute: y=-2x(-2x)2-2x2=6 Now find y-values!4x2-2x2=6 y=-2(3)y=-2(-3)2x2=6 y=-23y=63x2=3 x=±3 Solutions: 3, -23 and (-3, 63)3. Given fx=-2x+3 and gx=x2-6x+3, find the x-value(s) that satisfy fx=gx.fx=gx means they are equal when the graphs intersect! -2x+3=x2-6x+3 0=x2-4x Now find y-values or f(x)!0=x(x-4) y=-20+3y=-24+3x=0 and x=4 y=3y=-5Solutions: 0, 3 and (4, -5)4. Algebraically, determine the points of intersection of x-12+y-22=4 and y-2=2x.Substitute: y=2x+2 (x-1)2+(2x+2-2)2=4 x-1x-1+(2x)2=4 Quadratic Formula?x2-2x+1+4x2=4 Completing the Square?5x2-2x-3=0 AC Method: 5-3=-155x2-5x+3x-3=0-5, 35xx-1+3x-1=0 Grouping5x+3x-1=0 Now find y-values!x=-35 and 1 y-2=2-35y-2=2(1)y=1y=4Solutions: -35, 1 and 1, 45. Solve the following system of equatons algebraically:x+2y=0x2-2x+y2-2y-3=0Substitute: x=-2y-2y2-2-2y+y2-2y-3=0 4y2+4y+y2-2y-3=0 Completing the Square?5y2+2y-3=0 AC Method?y=-(2)±(2)2-4(5)(-3)2(5)Quadratic Formula: a=5, b=2, c=-3y=-2±4+6010=-2±6410 Now find x-values!y=-2±810 x+235=0x+2-1=0y=35 and -1 x=-65x=2Solutions: -65,35 and 2, -1Push Yourself! Applications of Systems of Equations DECIMALS use Calculator6. Sabrina is playing ball with her dog. She throws the ball in a parabolic path that can be modeled by the function y=-12x-32+7. Her brother, Bobby, is playing in a tree next to her. Bobby shines his laser pointer from the tree in a line that can be modeled by the function y=-12x+8.5. At what point(s) will the ball and the laser beam intersect?-12x-32+7=-12x+8.5 Homework Lesson 7.5: Solving Systems of Equations Algebraically1. Find all solutions of the system of equation algebraically:x-22+y+32=4x-y=32. Given fx=-2x+3 and gx=x2-6x+3, find the x-value(s) that satisfy fx=gx.3. Algebraically, determine the points of intersection of -y2+6y+x-9=0 and 6y=x+274. A boy standing on the top of a building in Albany throws a water balloon up vertically. The height, h (in feet) of the water balloon is given by the equation ht=-16t2+64t+192, where t is the time (in seconds) after he threw the water balloon. What is the value of t when the balloon hits the ground? Explain and show how you arrived at the answer.5. What is the total number of points of intersection of the graphs of the equations 2x2-y2=8 and y=x+2?(1) 1 (2) 2 (3) 3 (4) 06. Amy solved the equation 2x2+5x-42=0. She stated that the solutions to the equation were 72 and -6. Do you agree with Amy’s solutions? Explain why or why not. ................
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