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Algebra II - Quadratics in One Variable

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Study GuideAlgebra II–QuadraƟcs in One Variable1.QuadraƟc EquaƟonsAquadratic equationis a type of algebraic equation that includes a variable raised to the power of 2(that is, squared).The general or standard form of a quadratic equation is:Here is what each part means:•(x)is the variable you are solving for.•(a), (b), and (c)are numbers calledcoefficientsorconstants.•(a) cannot be zero, because if (a = 0), the (x2) term disappears and the equation is no longerquadratic.Quadratic equations appear often in algebra, physics, engineering, and many real-world problems.Because of this, it is important to know different ways to solve them.Common Methods to Solve Quadratic EquationsThere arefour main algebraic methodsused to solve quadratic equations. Each method works bestin different situations.1. FactoringFactoring means rewriting the quadratic expression as a product of two simpler expressions.Example idea:Once factored, you can set each factor equal to zero to find the solutions.

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Study Guide2. Square Root PropertyThis method is useful when the equation can be written in the form:To solve it, take the square root of both sides.Example idea:3. Completing the SquareCompleting the square rearranges the equation so one side becomes a perfect square.This method is helpful when factoring is difficult or impossible.It is also the idea used to derive the quadratic formula.4. Quadratic FormulaThequadratic formulais a universal method that works forany quadratic equation, even when theother methods do not work easily.Students often use this method when the equation cannot be factored.Summary•Aquadratic equationhas the form(ax2+ bx + c = 0)where(a≠0).•The equation contains asquared variable ((x2)), which makes it quadratic.•There arefour main methodsto solve quadratic equations:1.Factoring2.Square Root Property3.Completing the Square

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Study Guide2.Solving QuadraƟc by FactoringQuadratic equations can often be solved byfactoring. This method uses an important rule called theZero Product Property.The Zero Product PropertyTheZero Product Propertysays:If two numbers multiply to give0, thenat least one of them must be 0.In math form:Ifab = 0, thena = 0orb = 0.This idea is the key to solving quadratic equations by factoring.Example 1:ProblemSolve:(2x2=-9x-4)Let’s solve this step by step.Step 1: Move all terms to one sideTo factor a quadratic equation, we first need to write it in the form:Add (9x) and (4) to both sides:Step 2: Factor the quadraticNow we factor the expression:

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Study GuideIt factors into:Step 3: Apply the Zero Product PropertyIf the product of two expressions equals0, thenone of them must be 0.So we set each factor equal to zero:orStep 4: Solve each equationSolve the first equation:Now solve the second equation:Final SolutionsThese are the values ofxthat make the original equation true.

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Study GuideSummary•Factoringis a method used to solve quadratic equations.•It relies on theZero Product Property:•If (ab = 0), then (a = 0) or (b = 0).•Steps to solve by factoring:1.Move all terms to one side so the equation equals0.2.Factorthe quadratic expression.3.Set each factor equal to0.4.Solve the resulting equations.Example result:For (2x2+ 9x + 4 = 0)Solutions are:3.Quiz: Solving QuadraƟcs by Factoring1.QuestionSolve the equation:(x2+ 2x-8 = 0)Answer Choices-4, 2-2, 4-8, 1

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Study GuideCorrect Answer-4, 2Why This Is CorrectFactor the quadratic:Set each factor equal to zero:•(x + 4 = 0⇒x =-4)•(x-2 = 0⇒x = 2)So the solutions are-4 and 2.2.QuestionSolve the equation:(x2-11x =-30)Answer Choices-5, 6-6, 55, 6Correct Answer5, 6Why This Is CorrectMove (-30) to the left:

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Study GuideFactor:Solve:•(x = 5)•(x = 6)3.QuestionSolve the equation:(2x2= 28-10x)Answer Choices-7, 4-7, 2-4, 7Correct Answer-7, 2Why This Is CorrectMove everything to one side:Divide by 2:Factor:

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Study GuideSolutions:•(x =-7)•(x = 2)4.QuestionSolve the equation:(4x2= 25)Answer ChoicesCorrect AnswerWhy This Is CorrectDivide by 4:Take the square root:So the solutions are(-5/2) and (5/2).

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Study Guide5.QuestionSolve the equation:(6x2+ x-2 = 0)Answer ChoicesCorrect AnswerWhy This Is CorrectFactor the quadratic:Set each factor equal to zero:•(3x + 2 = 0⇒x =-(2/3)•(2x–1 = 0⇒x =(1/2)So the solutions are–(2/3) and(1/2).4.Solving QuadraƟcs by the Square Root PropertyAnother useful way to solve some quadratic equations is by using theSquare Root Property. Thismethod works best when the equation contains a squared term by itself, such as (x2) or ((x-a)2).The Square Root PropertyTheSquare Root Propertystates:x2= c⇒x =±√{c}This means that if a squared number equals a valueI, then the variable can be either thepositivesquare rootor thenegative square rootofI.•IfIis positive, the equation hastwo real solutions.•IfIis negative, the equation hastwo imaginary (complex) solutions.

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Study GuideExample1:Solve Each EquationLet’s solve each equation step by step.1.(x2= 48)Take the square root of both sides.Simplify the square root:Answer:2. (x2=-16)Take the square root of both sides:Since the square root of a negative number involves (i):Answer:3. (5x2–45 = 0)First isolate (x2).Add 45 to both sides:
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Document Details

University
Texas A&M University
Course
Algebra II
Subject
Mathematics

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