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Algebra 1Unit 3A: Factoring & Solving Quadratic EquationsNotesName: Block: Teacher:Algebra 1Unit 3A Notes:Quadratic Functions Factoring and SolvingQuadratic Functions andEquationsDISCLAIMER: We will be using this note packet for Unit 3A. You will be responsible for bringingthis packet to class EVERYDAY. If you lose it, you will have to print another one yourself. Anelectronic copy of this packet can be found on my class blog.1

Algebra 1Unit 3A: Factoring & Solving Quadratic EquationsStandardWrite expressions in equivalent forms to solve problemsMGSE9–12.A.SSE.3Choose and produce an equivalent form of an expression to reveal and explainproperties of the quantity represented by the expression.MGSE9–12.A.SSE.3aFactor any quadratic expression to reveal the zeros of the function defined by theexpression.MGSE9–12.A.SSE.3bComplete the square in a quadratic expression to reveal the maximum and minimumvalue of the function defined by the expression.Interpret structure of expressionsMGSE9‐12.A.SSE.2Use the structure of an expression to rewrite it in different equivalent forms. Forexample, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares thatcan be factored as (x2 – y2)(x2 y2).Create equations that describe numbers or relationshipsMGSE9–12.A.CED.1Create equations and inequalities in one variable and use them to solve problems.Include equations arising from quadratic functions.MGSE9-12.A.CED.2Create quadratic equations in two or more variables to represent relationshipsbetween quantities; graph equations on coordinate axes with labels and scales. (Thephrase “in two or more variables” refers to formulas like the compound interestformula, in which A P(1 r/n)nt has multiple variables.)MGSE9–12.A.CED.4Rearrange formulas to highlight a quantity of interest using the same reasoning as insolving equations. Example: Rearrange area of a circle formula A π r2 to highlightthe radius r.Solve equations and inequalities in one variableMGSE9‐12.A.REI.4Solve quadratic equations in one variable.MGSE9–12.A.REI.4aUse the method of completing the square to transform any quadratic equation in xinto an equation of the form (x – p)2 q that has the same solutions. Derive thequadratic formula from ax2 bx c 0.MGSE9–12.A.REI.4bSolve quadratic equations by inspection (e.g., for x2 49), taking square roots,factoring, completing the square, and the quadratic formula, as appropriate to theinitial form of the equation (limit to real number solutions).NotesLesson2

Algebra 1Unit 3A: Factoring & Solving Quadratic EquationsNotesUnit 3A: Factoring & Solving Quadratic EquationsTable of ContentsAfter completion of this unit, you will be able to Learning Target #1: Factoring Factor the GCF out of a polynomial Factor a polynomial when a 1 Factor a polynomial when a 1 Factor special products (difference of two squares)LessonPageDay 1 - Factoring QuadraticExpressions – GCF4Day 2 - Factoring QuadraticTrinomials, a 16Day 3 - Factoring QuadraticTrinomials, a 18Day 4 - Factoring SpecialProducts10Learning Target #3: Solving by Non Factoring Methods Solve a quadratic equation by finding square roots. Solve a quadratic equation by completing the square. Solve a quadratic equation by using the QuadraticFormula.Day 5 – Solving Quadratics(GCF, a 1, a 1)12Day 6 – SolvingBy Taking Square Roots16Learning Target #4: Solving Quadratic Equations Solve a quadratic equation by analyzing the equation anddetermining the best method for solving. Solve quadratic applicationsDay 7 - Solving by Completingthe Square18Day 8 - Solving by QuadraticFormula20Learning Target #2: Solving by Factoring Methods Solve a quadratic equation by factoring a GCF. Solve a quadratic equation by factoring when a is not 1. Create a quadratic equation given a graph or the zeros ofa function.Timeline for Unit 3AMondayJanuary 27thTuesdayJanuary 28thDay 1- FactoringQuadraticExpressions – GCFWednesday29thDay 2 - FactoringQuadraticTrinomials, a 1Thursday30thDay 3 - FactoringQuadraticTrinomials, a 1February 3rdDay 5a – SolvingQuadratics(GCF, a 1, a 1)4thDay 5b – SolvingQuadratics(GCF, a 1, a 1)5thDay 6a – SolvingBy Taking SquareRoots6thDay 6b – SolvingBy Taking SquareRoots10th11th12thReview SolvingQuadraticsQuiz – SolvingQuadratics13thDay 7b Solving byCompleting theSquareDay 8 Solving byQuadratic FormulaUnit 3A Test ReviewFriday31stDay 4 - FactoringSpecial ProductsQuiz – FactoringQuadratics7thDay 7a Solving byCompleting theSquare14thUnit 3A Test3

Algebra 1Unit 3A: Factoring & Solving Quadratic EquationsNotesDay 1 – Factor by GCFStandard(s):MGSE9–12.A.SSE.3a Factor any quadratic expression to reveal the zeros of the function definedby the expression.What is Factoring?Factoring Finding out which two expressions you together to get one single expression. “Splitting” an expression into a product of simpler expressions. The opposite of expanding or distributing.Numbers have factors:Expressions have factors too:Review: Finding the GCF of Two NumbersCommon Factors Factors that are shared by two or more numbersGreatest Common Factor (GCF) To find the GCF create a factor t-chart for each number and find the largest common factorExample: Find the GCF of 56 and 104Practice: Find the GCF of the following numbers.a. 30, 45b. 12, 544

Algebra 1Unit 3A: Factoring & Solving Quadratic EquationsNotesFinding the GCF of Two ExpressionsTo find the GCF of two expressions, create a factor chart for the two numbers AND expand thevariables. Circle what is common to both.Example: Find the GCF of 36x2y and 16xyPractice: Find the GCF of the following pairs of expressions.1) 15x3 and 9x22) 9a2b2, 6ab3, and 12b3) 8x2 and 7y3Factoring by GCFSteps for Factoring by GCF1. Find the greatest common factor of all the terms.2. The GCF of the terms goes on the outside of the expression and what is leftover goes inparenthesis after the GCF.3. After “factoring out” the GCF, the only that number that divides into each term should be 1.Practice: Factor each expression.1) x2 5xGCF 2)x2 – 8x4)18x2 – 6x5)-2m2 – 8m7)6x3 – 9x2 12x8)4x3 6x2 – 8xGCF GCF GCF GCF GCF 3)28x - 63GCF 6)-9a2 - aGCF 9) 15x3y2 10x2y4GCF 5

Algebra 1Unit 3A: Factoring & Solving Quadratic EquationsNotesDay 2 – Factor Trinomials when a 1Standard(s):MGSE9–12.A.SSE.3a Factor any quadratic expression to reveal the zeros of the function definedby the expression.2nd Degree(Quadratic)Quadratic Trinomials3 Terms(Trinomial)ax 2 bx cFactoring a trinomial means finding two that when multiplied togetherproduce the given trinomial.Skill Preview: “Big X” ProblemsComplete the diamond problems. The top cell contains the product of the numbers in the left andright cells, while the bottom cell contains the sum.6

Algebra 1Unit 3A: Factoring & Solving Quadratic EquationsNotesFactoring using Quadratic Trinomials when a 1Steps for Factoring when a 1Step 1: ALWAYS check to see if you can factor out a GCFStep 2: Draw parentheses for the binomial factors and fill in the variables. Ex ( x)(x)Step 3: Complete a “Big X” and T-chartStep 4: Determine what two numbers can be multiplied to get your “a c” term and added to getyour “b” term. (Use a factor T-chart)Step 5: Fill the factors in the parentheses.Example: Factor the trinomial.x2 6x 8factored form:Factor the following trinomials.a. Factor x2 4x – 32c. Factor x2 – 36b. Factor x2 – 3x – 18d. Factor 2x2 16x 24Remember: You must ALWAYS include the GCF on the outside of the factored form!7

Algebra 1Unit 3A: Factoring & Solving Quadratic EquationsNotesDay 3 – Factor Trinomials with a 1In the previous lesson, we factored polynomials for which the coefficient of the squared term, “a”was always 1. Today we will focus on examples for which a 1.Looking for PatternsWhat do you observe in the following Area Models?Factoring is theofdistributing or multiplying.STEP 1: ALWAYS check to see if you can factor out a GCF.Factor: 2𝑥 2 5𝑥 3STEP 2: Complete a “Big X” and T-chart Determine what two numbers can be multiplied toget your “a c” term and added to get your “b”term.STEP 3: Create a 2x2 Area Model and place your original“a” term in the top left box and “c” term in thebottom right box. Fill the remaining two boxes with the two numbersyou found in “Big X” and place an x after them.STEP 4: Factor out a GCF from each row and columnSTEP 5: Check your factors on the outside by multiplyingthem together to make sure you get all theexpressions in your box.Factored Form:8

Algebra 1Unit 3A: Factoring & Solving Quadratic EquationsNotesFactoring a 1Using the Area Model. Factor the following trinomials.1. 5𝑥 2 14𝑥 3Factored Form:2. 2𝑥 2 17𝑥 30Factored Form:3. 12x2 56x 64Factored Form:4. 6x2 – 40x 24Factored Form:Remember: You must ALWAYS include the GCF on the outside of the factored form!9

Algebra 1Unit 3A: Factoring & Solving Quadratic EquationsNotesDay 4 – Factor Special ProductsStandard(s): MGSE9‐12.A.SSE.2Use the structure of an expression to rewrite it in different equivalent forms. For example, see x4 – y4as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 y2).Review: Factor the following expressions:a. x2 – 49b. x2 – 25c. x2 – 811. What do you notice about the “a” term?2. What do you notice about the “c” term?3. What do you notice about the “b” term?4. What do you notice about the factored form?The above polynomials are a special pattern type of polynomials; this pattern is called aDifference of Two Squaresa2 – b2 (a – b)(a b)*Always subtraction**Both terms are perfect squares**Always two terms*Can you apply the “Difference of Two Squares” to the following polynomials?a. 9x2 – 49b. 9x2 – 100c. 4x2 – 25d. 16x2 – 1e. x2 25f. 25x2 – 64g. 36x2 – 81h. 49x2 – 910

Algebra 1Unit 3A: Factoring & Solving Quadratic EquationsReview: Factor the following expressions:a. x2 8x 16b. x2 – 2x 1Notesc. x2 – 10x 251. What do you notice about the “a” term?2. What do you notice about the “c” term?3. What do you notice about the “b” term?4. What do you notice about the factored form?The above polynomials are a second type of pattern; this pattern type is called aPerfect Square Trinomialsa2 2ab b2 (a b)2a2 – 2ab b2 (a – b)2Using the perfect square trinomial pattern, see if you can fill in the blanks below:a. x2 36b. x2 - 81c. x2 - 64d. x2 4x e. x2 – 6x f. x2 20x 11

Algebra 1Unit 3A: Factoring & Solving Quadratic EquationsNotesDay 5 – Solving Quadratics (GCF, when a 1, when a not 1)Standard(s):MGSE9‐12.A.REI.4 Solve quadratic equations in one variable.The Main Characteristics of a Quadratic Function A quadratic function always has an exponent of . The standard form of a quadratic equation is The U-shaped graph is called a . The highest or lowest point on the graph is called the . The points where the graph crosses the x-axis are called the. The points where the graph crosses are also called theto the quadratic equation. Aquadratic equation can have , , orsolutions.In this unit, we are going to explore how to solve quadratic equations.Solving a quadratic equation really means:Finding its , , or .Create an equation to represent the following graphs:Zeros: x &Zeros: x &y y 12