Solving Quadratics Notes - Commack Schools
Common CoreAlgebra 2Chapter 3:Quadratic Equations &Complex Numbers1
Chapter Summary: The strategies presented for solving quadratic equations in this chapter were introduced at the end ofAlgebra. The difference now is that solutions are not restricted to real numbers. In Section 3.2, complex numbers are defined and operations on complex numbers will be presented. Thisif followed by the technique of completing the square so that the Quadratic formula can be derived. In total, we will use five strategies for solving quadratic equations: graphing, square rooting, factoring,completing the square, and using the Quadratic Formula. The last two sections extend work with solving quadratic equations to solving nonlinear systems andsolving quadratic inequalities. Each of these topics requires recall of connected skills from work with linearequations. Nonlinear systems are solved by methods of graphing, substitution, and elimination.2
Section 3.1: Solving Quadratic EquationsEssential Question: How can you use the graph of a quadratic equation to determine the number of realsolutions of equation?What You Will Learn Solve quadratic equations by graphing. Solve quadratic equations algebraically. Solve real-life ----Solving Quadratic Equations by GraphingA quadratic equation in one variable is an equation that can be written in the standard form:ππ₯ 2 ππ₯ π 0, where π, π, and π are real numbers and π 0.A root of an equation is a solution of the equation. We can use various methods to solve quadratic equations.When solving a quadratic equation we are looking for all the possible values of π₯ that make the equation Example 1: Solve each equation by graphing.a) π₯ 2 π₯ 6 0b) 2π₯ 2 2 4π₯3
Solving Quadratic Equations Algebraically using Square RootsWhen solving quadratic equations using square roots, you can use properties of square roots to write yoursolutions in different forms.Example 2: Solve each equation using square roots.a) 3π₯ 2 9 0b) 4π₯ 2 31 49c)52(π₯ 3)2 ing a Quadratic Equations Algebraically by FactoringWhen the left side of ππ₯ 2 ππ₯ π 0 is factorable, we can solve the equation using the:Zero-Product Property.Example 3: Solve the following equations by factoring.a) π₯ 2 4π₯ 45b) 2π₯ 2 11π₯ 12 04c) π₯ 2 8π₯
Zeros of a Quadratic FunctionWe know the x-intercepts of the graph of π(π₯) π(π₯ π)(π₯ π) are and . The value of thefunction is equal to when π₯ π and π₯ π, therefore the numbers π and π are called π§ππππ ofthe function.A zero of a function π is an π₯-value for which π(π₯) 0.Example 4: Find the zero(s) of each of the functions.a) π(π₯) π₯ 2 12π₯ 35b) π(π₯) 3π₯ 2 5π₯d) π(π₯) 4π₯ 2 28π₯ 49e) π(π₯) 3π₯ 2 95c) π(π₯) π₯ 2 2π₯ 3
Solving Real-Life ProblemsWhen an object is dropped, its height β (in feet) above the ground after π‘ seconds can be modeled by thefunctions β 16π‘ 2 β0 , where β0 is the initial height (in feet) of the object.The graph of β 16π‘ 2 200, representing the height of an objectdropped from an initial height of 200 feet, is shown to the left.How long does it take for this object to hit the ground?The model β 16π‘ 2 β0 assumes that the force of air resistance onthe object is negligible. Also, this model applies only to objects droppedon Earth. For planets with stronger or weaker gravitational forces,different models are used.Example 5: Modeling a Dropped ObjectFor a science competition, students must design a container that prevents an egg from breaking whendropped from a height of 50 feet.a) Write a function that gives the height β (in feet) of the container after π‘ seconds.b) How long does it take for the container to hit the ground?c) Find and interpret β(1) β(1.5).6
Section 3.2: Complex NumbersEssential Question: What are the subsets of the set of complex numbers?In your study of mathematics, you have probably worked with only ππππ ππ’πππππ , which can be representedgraphically on the real number line.In this lesson, the system of numbers is expanded to include πππππππππ¦ ππ’πππππ . The set of real numbersand imaginary numbers make up the set of πππππππ₯ ππ’πππππ .What You Will Learn Define and use the imaginary unit π Add, subtract, and multiply complex numbers. Find complex solutions and --THE COMPLEX NUMBER ------------------------------Classifying NumbersExample 1: Determine which subsets of the set of complex numbers.a) 9b) 04d) 9c) 47e) 1
The Imaginary Unit πNot all quadratic equations have real-number solutions. For example, π₯ 2 3 has no real-number solutionssince the square of any real number is never a negative number.To overcome this problem, mathematicians created an expanded system of numbers using the imaginaryunit π, defined as π 1. The imaginary unit π can be used to write the square root of any negative number.Remember:π 1π 2 ing Square Roots of Negative NumbersExample 2: Find the square root of each number.a) 25b) 5 9e) 32f) Simplify: 25 49 48 75c) 728d) 2 54
Complex NumbersA complex number written in π π‘ππππππ ππππ is a number π ππ where π and π are real numbers.π ππIf π 0, then π ππ is an imaginary number.If π 0 and π 0, then π ππ is a pure imaginary number.The diagram to the right shows how different types ofcomplex numbers are ---Equality of Two Complex NumbersExample 3: Find the values of π₯ and π¦ that satisfy each of the equations.a) 2π₯ 7π 10 π¦πb) π₯ 3π 9 π¦πc) 9 4π¦π 2π₯ 3πOperations with Complex NumbersExample 4: Add or subtract. Write the answer in standard form.a) (8 π) (5 4π)b) (7 6π) (3 6π)9c) 13 (2 7π) 5π
Example 5: Multiply. Write the answer in standard form.a) 4π( 6 π)b) (9 2π)( 4 7π)c) ( 3 2π)( 3 -Complex Solutions and ZerosExample 6: Solve the following quadratic equations.a) π₯ 2 4 0b) 2π₯ 2 11 47c) HOW DO YOU SEE IT? The graphs of three functions are shown. Which function(s) has real zeros? Whichfunction(s) have imaginary roots? Explain your reasoning.10
Section 3.3: Completing the SquareEssential Question: How can you complete the square for a quadratic expression?Completing the Square A process used by adding a term π to an expression of the form π₯ 2 ππ₯ such thatπ₯ 2 ππ₯ π is a perfect square trinomial.Value of π needed toβcomplete the squareβExpressionExpression written as abinomial squared1) π₯ 2 2π₯ π2) π₯ 2 4π₯ π3) π₯ 2 8π₯ π4) π₯ 2 10π₯ ππ₯211π₯π₯11Look for a pattern in the middle column of the table.How are π and π related?Rule:Look for patterns in the last column of the table. Consider the general statement π₯ 2 ππ₯ π (π₯ π)2 .How are π and π related?Rule:11
What You Will Learn Solve quadratic equations using square roots. Solve quadratic equations by completing the square. Write quadratic functions in vertex Solving Quadratic EquationsExample 1: Solve the following quadratic equations.a) (π₯ 8)2 100b) (π₯ 4)2 25c) 2(π₯ 7)2 40Solving Quadratic Equations by Completing the Square when [π π]Example 2: Solving the following quadratic equations by completing the square. Then identify the vertex.a) π₯ 2 10π₯ 7 0b) π₯ 2 8π₯ 4 012
c) π₯ 2 5π₯ 1 0d) π₯ 2 3π₯ 11 ing Quadratic Equations by Completing the Square [when π π]Example 3: Solve the following quadratic equation by completing the square. Then identify the vertex.a) 3π₯ 2 12π₯ 15 0b) 4π₯ 2 24π₯ 11 013
Modeling with MathematicsExample 4: The height π¦ (in feet) of a baseball π‘ seconds after David Wrighthits the ball can be modeled by the function:π¦ 16π‘ 2 96π‘ 3Find the maximum height of the baseball.METHOD #1METHOD #2How long does the ball take to hit the ground?14
Name DateChapter3QuizFor use after Section 3.3Solve the equation by using the graph. Check your solution(s).1. x 2 x 12 0Answers3. x 2 5 x 62. 2 x 2 4 7 x1.2.3.4.5.6.7.Solve the equation using square roots or by factoring.4. x 2 3 x 1 5. x x 1 3 6. 2 x 2 2 x 57. Find the values of x and y that satisfy the equation 5 x 8i 30 yi.8.9.10.Perform the operation. Write your answer in standard form.8. 3 4i 6 2i 79. 6i 4 3i 11.12.10. Find the zeros of the function f ( x ) 5 x 2.213.Solve the equation by completing the square.11. x 2 16 x 22 012. x 2 12 x 26 014. a.13. Write y x 2 4 x 5 in vertex form. Then identify the vertex.b.14. A water balloon is tossed into the air so that its height h (in feet) afterc.t seconds can be modeled by the function h t 16t 2 80t 5.a. What is the height of the balloon after 1 second?b. For how long is the balloon more than 30 feet high?c. What is the maximum height of the balloon?15. A rectangular lawn measuring 24 feet by 16 feet is surrounded by aflower bed of uniform width. The combined area of the lawn and theflower bed is 660 square feet. What is the width of the flower bed?1515.
Section 3.4 β Using the Quadratic FormulaEssential Question: How can you derive a general formula for solving a quadratic equation?What You Will Learn Solve quadratic equations using the Quadratic Formula. Analyze the discriminant to determine the number and type of solutions. Solve real-life ----DERIVATION OF THE QUADRATIC FORMULALetβs begin with a quadratic equation in standard form:16
Solving Equations Using the Quadratic FormulaPreviously, you solved quadratic equations by completing the square. We developed a formula that gives thesolutions of any quadratic equation by completing the square for the general equation ππ₯ 2 ππ₯ π 0.The formula for the solutions is called the QUADRATIC FORMULA.Solving an Equation with Two Real SolutionsExample 1: Solve π₯ 2 3π₯ 5 using the Quadratic Formula.SKETCHExample 2: Solve the equation using the Quadratic Formula.(a) π₯ 2 6π₯ 4 0(b) 2π₯ 2 4 7π₯17(c) 5π₯ 2 π₯ 8
Solving an Equation with One Real SolutionExample 3: Solve 25π₯ 2 8π₯ 12π₯ 4 using the Quadratic Formula.SKETCHSolving an Equation with Imaginary SolutionsExample 4: Solve π₯ 2 4π₯ 13 using the Quadratic Formula.SKETCHExample 5: Solve the following quadratic equations.(a) π₯ 2 41 8π₯(b) 9π₯ 2 30π₯ 2518(c) 5π₯ 7π₯ 2 3π₯ 4
ANALYZING THE DISCRIMINANTIn the Quadratic Formula, the expression:π2 4ππis called the discriminant of the equation ππ₯ 2 ππ₯ π 0. π π2 4πππ₯ 2πWe can analyze the discriminant of a quadratic equation to determine the number and type of solutions of theequation.Example 6: Find the discriminant of the quadratic equation and describe the number and type of solutions ofthe equation.(a) 4π₯ 2 8π₯ 4 0(b)12π₯2 π₯ 1 0(c) 5π₯ 2 8π₯ 1319
(d) 7π₯ 2 3π₯ 6(e) 4π₯ 2 6π₯ 9(f) 5π₯ 2 1 6 -Example 7: Writing an EquationFind a possible pair of integer values of π and π so that the equation ππ₯ 2 4π₯ π 0 has one real solution.The write the equation.Find a possible pair of integer values for π and π so that the equation ππ₯ 2 3π₯ π 0 has two realsolutions. Then write the equation.20
Solving Real Life ProblemsThe function β 16π‘ 2 β0 is used to model the height of a dropped object. For an object that is launchedor thrown, and extra term π£0 π‘ must be added to the model to account for the objectβs initial vertical velocity,π£0 (in feet per second).Recall that β is the height (in feet), π‘ is the time in motion (in seconds), and β0 is the initial height (in feet).β 16π‘ 2 β0 β 16π‘ 2 π£0 π‘ β0 As shown below, the value of π£0 can be , , or dependingon whether the object is launched upward, downward, or parallel to the ground.21
Example 8: A juggler tosses a ball into the air. The ball leaves the jugglerβs hand 4 feet above the ground andhas an initial vertical velocity of 30 feet per second. The juggler catches the ball when it falls backto a height of 3 feet. How long is the ball in the air?Example 9: A lacrosse player throws a ball in the air from an initial height of 7 feet. The ball has an initialvertical velocity of 90 feet per second. Another player catches the ball when it is 3 feet above theground. How long is the ball in the air?Example 10: In a volleyball game, a player on one team spikes the ball over the net when the ball is 10 feetabove the court. The spike drives the ball downward with an initial vertical velocity of 55 feetper second. How much time does the opposing team have to return the ball before it touchesthe court?22
Section 3.5 β Solving Nonlinear SystemsEssential Question: How can you solve a nonlinear system of equations?What You Will Learn Solve systems of nonlinear equations. Solve quadratic equations by ----Systems of Nonlinear EquationsPreviously, weβve solved systems of linear equations by graphing, substitution, and elimination. You can alsouse these methods to solve a system of nonlinear equations.In a system of nonlinear equations, at least one of the equations in nonlinear. For instance, the nonlinearsystem shown below has a equation and a equation.π¦ π₯ 2 2π₯ 4π¦ 2π₯ 5When the graphs of the equations in a system are a line and a parabola, the graphs can intersect in zero, one,or two points. So the system can have zero, one, or two solutions.Example 1:Solve the above system by graphing.Example 2:Solve the above system by substitution.d23
When the graphs of the equations in a system are a parabola that opens up and parabola that opens down,the graphs can intersect in zero, one, or two points. So, the system can have zero, one, or two solutions.Example 3: Solving the following system by elimination.π¦ 2π₯ 2 5π₯ 2π¦ π₯ 2 2π₯Example 4: Solve the following equations by any method.(a) π¦ π₯ 2 6π₯ 15π¦ (π₯ 3)2 6(c) π¦ (π₯ 4)(π₯ 1)π¦ π₯ 2 3π₯ 424
Some nonlinear systems have equations of the form: π₯ 2 π¦ 2 π 2 . This equation is the standard form of acircle with center (0, 0) and radius π.When the graphs of the equations in a system are a line and a circle, the graphs can intersect in zero, one, ortwo points. So, the system can have zero, one, or two solutions.Example 5: Solve the following systems of equations.(a) π₯ 2 π¦ 2 10(b) π₯ 2 π¦ 2 11π¦ 3π₯ 101π¦ 2π₯ 225
Section 3.6: Quadratic InequalitiesEssential Question: How can you solve a quadratic inequality?What You Will Learn Graph quadratic inequalities in two variables. Solve quadratic inequalities in one ----Exploration: Solving a Quadratic InequalityThe graphing calculator screen shows the graph of: π(π₯) π₯ 2 2π₯ 3.Explain how you can use the graph to solve the inequality:π₯ 2 2π₯ 3 0Use the above graph to solve each of the following inequalities.(a) π₯ 2 2π₯ 3 0(b) π₯ 2 2π₯ 3 026(c) π₯ 2 2π₯ 3 0
Graphing Quadratic Inequalities in Two VariablesA quadratic inequality in two variables can be written in one of the following forms, where π, π, and π arereal numbers and π 0.π¦ ππ₯ 2 ππ₯ ππ¦ ππ₯ 2 ππ₯ ππ¦ ππ₯ 2 ππ₯ ππ¦ ππ₯ 2 ππ₯ πExample 1: Graph the following inequalities.π¦ π₯ 2 2π₯ 1π¦ π₯ 2 4π₯ 3π¦ π₯2 4π¦ π₯ 2 3π₯ 227
Solving Inequalities by GraphingExample 2: Using the graph of π(π₯) π₯ 2 3π₯ 10, solve each of the following inequalities.(a) π(π₯) 0(b) π(π₯) 0(c) π(π₯) 0(d) π(π₯) 0(e) π(π₯) 6(f) π(π₯) 6Using a Quadratic Inequality in Real LifeExample 3: A manila rope used for rappelling down a cliff can safely support a weight π (in pounds) provided:π 1480π 2where π is the diameter (in inches) of the rope. Graph the inequality and interpret the solution.28
Modeling with MathematicsExample 4:A rectangular parking lot must have a perimeter of 440 feet and an area of at least 8000 square feet. Describethe possible lengths of the parking lot.(a) Write an equation to represent the perimeter of the parking lot.(b) Write an inequality to represent the area of the parking lot.What can we do with the above equation and inequality to help us find all possible lengths?29
Review for Chapter 3 Test (Part 2)Solve the equation using any method. Provide a reason for your choice.1. 0 π₯ 2 2π₯ 32. 6π₯ π₯ 2 73. π₯ 2 49 854. (π₯ 4)(π₯ 1) π₯ 2 3π₯ 4Example how to use the graph to find the number and type of solutions of the quadratic equation. Justifyyour answer by using the discriminant.5.6.7.30
Solve the system of equations.8. π₯ 2 66 16π₯ π¦2π₯ π¦ 189. 0 π₯ 2 π¦ 2 40π¦ π₯ 410. Write (3 4π)(4 6π) as a complex number in standard form.11. The aspect ratio of a widescreen TV is the ratio of the screenβs width to its height, or 16:9. What are thewidth and the height of a 32-inch widescreen TV? [Hint: Use Pythagorean Theorem]12. The shape of the Gateway Arch in St. Louis, Missouri, can be modeled by π¦ 0.0063π₯ 2 4π₯, where π₯ isthe distance (in feet) from the left foot of the arch and π¦ is the height (in feet) of the arch above theground. For what distances π₯ is the arch more than 200 feet above the ground?31
In total, we will use five strategies for solving quadratic equations: graphing, square rooting, factoring, completing the square, and using the Quadratic Formula. The last two sections extend work with solving quadratic equations to solving nonlinear systems and solving quadratic inequalities.
QUADRATICS UNIT Solving Quadratic Inequalities and Curve Fitting By graphing the inequality: y x2 β 7x 10, we can begin to look at what shading would look like: Looking at the inequality: y β3x2 β 6x β 7, write the solution: We can also use our knowledge of and and or to solve much faster: x2 12x 39 12 x2 β 24 5x or and
3 Notes Solving Quadratics with Imaginary Numbers.notebook 1 January 11, 2017 Jan 4 9:06 AM Quadratic Functions MGSE9 12.N.CN.7 Solve quadratic equations with real coefficients that have complex solutions by (but not limited to) square
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Unit 2-1 Factoring and Solving Quadratics Learning Targets: Factoring Quadratic Expressions 1. I can factor using GCF. 2. I can factor by grouping. 3. I can factor when a is one. 4. I can factor when a is not equal to one. 5. I can factor perfect square trinomials. 6. I can factor using difference of squares. Solving Quadratic Equations 7.
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
Graphing-quadratics-worksheet-with-answers. SOLVING QUADRATIC EQUATIONS BY FACTORING WORKSHEET ANSWERS. SOLVING LINEAR. INEQUALITIES WORKSHEET KUTA POLYNOMIAL. Mar 19, 2021 β Worksheet Quadratic Equations solve Quadratic Equations by Peting from worksheet graphing quadratics from standard form answer . Aug 29, 2020 β Graphing quadratic
Nov 01, 2015Β Β· Solving Quadratics by Factoring β Part 1 Solve a quadratic equation by factoring: Once a quadratic equation is factored, we can use the zero product property to solve the equation. ! The zero product property states that if the product of two factors is
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