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Warm-UpSolving Quadratic Equations byFactoringReviewing Factoring SkillsFACTORING PERFECT SQUARESExample: Factor 16𝑥 2 8𝑥 1.Since the quadratic is in the form 𝑎2 2𝑎𝑏 𝑏 2 , factor using perfect squares.(4𝑥 1)(4𝑥 1) )2(FACTORING QUADRATICS WHEN 𝒂 𝟏Example: Factor 𝑥 2 𝑥 30.Since 𝑎 1, only worry about factors of –30. The factors of –30 that add upto –1 are –6 and 5.(𝑥 )(𝑥 )FACTORING DIFFERENCE OF SQUARESExample: Factor 9𝑥 2 49.()2 ((3𝑥 7)( Edgenuity, Inc.)2)1 1

Solving Quadratic Equations byFactoringWarm-UpReviewing Factoring SkillsFACTORING QUADRATICS WHEN 𝒂 𝟏Example: Factor 3𝑥 2 17𝑥 6.Since 𝑎 3 is prime, the factors of the first terms are 3𝑥 and 1𝑥. Next, find thefactors of –6, that when combined, the sum of the outer and inner values willbe 17.(3𝑥 )(1𝑥 )Lesson ObjectivesBy the end of this lesson, you should be able to: Findsolutions for quadratic equations using theproperty. Use key attributes of afunction to solve wordproblems.W2KWords to KnowFill in this table as you work through the lesson. You may also use the glossaryto help you.zero productproperty Edgenuity, Inc.the property stating that if athen at least one of theis equal to zero,must be zero2 2

Solving Quadratic Equations ic Functions and Their SolutionsA quadratic function can be written as: 𝑦, where 𝑎 0. The equation of the function graphed below is 𝑦 𝑥 2 𝑥 12. It can also be written in factored form as 𝑦 (𝑥 3)(𝑥 4). Factored form can help us identify the zeroes, or where the parabolacrosses the 𝑥-axis. The zeroes in the graph are –3 and 4. This is where 𝑓(𝑥) .Circle the 𝑥-intercepts.yy x 3 (x 4)5(4, 0)(–3, 0)–10–55x10–5–10 Edgenuity, Inc.3

Solving Quadratic Equations byFactoringInstructionSlide2Quadratic Functions and Their SolutionsThere are three situations in which you can have zeroes:1. Two zeroes – Parabola crosses the 𝑥-axis at two places.2. One zero – Parabola has vertex on the 𝑥-axis.zeroes – One example is a parabola with a vertex below the3.𝑥-axis and opening downward.The Zero Product PropertyExample: Solve 𝑥 2 𝑥 30 0.Zero product property: The property states that if a product is equal to zero, thenat least one of the factors must be zero. If 𝑎𝑏 0, thenor.Strategy: Factor, set each factor equal to 0, and solve.Step 1:Step 2:𝑥 6 𝑥 5 0𝑥 6 0 6 6𝑥 𝑥 5 0 5or 5𝑥 The solutions are 𝑥 6 or 𝑥 5. Edgenuity, Inc.4

InstructionSolving Quadratic Equations byFactoringSlide4Checking the SolutionsExample: Solve 3𝑥 2 17𝑥 6 0.yx3𝑥 1 0 and 𝑥 6 0–1010–5𝑥 and 𝑥 –15Check: Substitute each value for 𝑥 in the givenequation and simplify.1Substitute 𝑥 :3–252133 171 6 03117 18 093331 17 18 03 3318 18 033 0Substitute 𝑥 6:3 62 17 6 6 03 36 102 6 0108 102 6 06 6 0 0 Edgenuity, Inc.5

Solving Quadratic Equations byFactoringInstructionSlide7Using Quadratics to Model and Solve ProblemsExample: When the product of 8 and the square of a number is decreased by 5,the result is the product of –18 and the number. What could the number be? Write an equation.Let 𝑥 the unknown number. 18𝑥 Solve.Use the zero product property to find two solutions.8𝑥 2 18𝑥 5 0( 1)((4𝑥 1) 04𝑥 1 5) 0or(2𝑥 5) 02𝑥 5𝑥 𝑥 Answer the problem.The number could be 𝑥 Edgenuity, Inc.15or 𝑥 .426

InstructionSolving Quadratic Equations byFactoringSlide7Using Quadratics to Model and Solve Problems Check.Substitute each value for 𝑥 in the original equation and simplify.1Substitute 𝑥 :42118 5 184411018 162481 109 2 22 The left side equals the right side, so5Substitute 𝑥 :258 2892is a correct answer.2 5 18 522590 5 4250 5 4545 45Both numbers are answers to the question. Edgenuity, Inc.7

InstructionSolving Quadratic Equations byFactoringSlide9Using Quadratics to Model and Solve ProblemsA man throws a ball into the air with an initial velocity of 50 feet per second and aninitial height of 6 feet. The graph shows the height of the ball over time.𝑦 16𝑥 2 50𝑥 6y60height 454020initial6–10 3 sec10xball hits theground Edgenuity, Inc.8

Summary?LessonQuestionSolving Quadratic Equations byFactoringHow can the unique multiplication property of zero be used to solvea quadratic equation?AnswerSlide2Review: Key ConceptsZero product property: If 𝑎𝑏 0, then 𝑎 or 𝑏 .yx 3 x 4 05(–3, 0)(4, 0)–10x10–5Use theproperty to solve a quadratic equation inform, set equal to 0:𝑎𝑥 2 𝑏𝑥 𝑐 0Solutions are the Edgenuity, Inc.-intercepts of the graph.9

Solving Quadratic Equations byFactoringSummarySlide2Review: Common Problem TypesTo solve a quadratic equation by factoring:1. Write the equation in standard form, so that one side is equal to.2.the expression.3. Use the zero product property to write and solve linear equations.4. Check the solutions.To solve a quadratic word problem:1. Write aequation to model the word problem.2. Solve the equation.3. Answer the problem, then check the solution in Edgenuity, Inc.10

SummarySolving Quadratic Equations byFactoringUse this space to write any questions or thoughts about this lesson. Edgenuity, Inc.11

Factoring Review: Common Problem Types To solve a quadratic equation by factoring: 1. Write the equation in standard form, so that one side is equal to. 2. the expression. 3. Use the zero product property to write and solve linear equations. 4. Check the solutions. To solve a quadratic word problem: 1.