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FOA/Algebra 1Unit 3A: Factoring & Solving Quadratic EquationsNotesDay 1 – Factor by GCFStandard(s):What is Factoring?Factoring Finding out which two expressions you multiplied together to get onesingle expression.Is like “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 numbers are called common factors.Greatest Common Factor (GCF) The greatest of the common factors is called the Greatest Common Factor (GCF). To find the greatest common factor, you can make a factor tree and complete the primefactorization of both numbers. The GCF is the product of the common prime factors. You can also do a factor t-chart for each number and find the largest common factor Example: Find the GCF of 56 and 1042

FOA/Algebra 1Unit 3A: Factoring & Solving Quadratic EquationsPractice: Find the GCF of the following numbers.a. 30, 45b. 12, 54NotesFinding the GCF of Two ExpressionsTo find the GCF of two expressions, you will complete the prime factorization of the two numbers orfactor chart of the two numbers AND expand the variables. Circle what is common to both.Example: Find the GCF of 36x2y and 16xyPractice: Find the GCF of the following pairs of expressions.1) 100 and 602) 15x3 and 9x23) 9a2b2, 6ab3, and 12b4) 8x2 and 7y33

FOA/Algebra 1Unit 3A: Factoring & Solving Quadratic EquationsNotesFactoring 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. 28x - 63GCF 5. 18x2 – 6x7. 2m2 – 8mGCF 8. -9a2 - a10. 6x3 – 9x2 12xGCF GCF GCF GCF 11. 4x3 6x2 – 8xGCF 3. x2 – 3xGCF 6. 4x2 – 4xGCF 9. 35y2 – 5yGCF 12. 15x3y2 10x2y4GCF 4

FOA/Algebra 1Unit 3A: Factoring & Solving Quadratic EquationsNotesDay 2 – Factor Trinomials when a 1Standard(s):2nd Degree(Quadratic)Quadratic Trinomials2ax bx c3 Terms(Trinomial)Factoring a trinomial means finding two binomials that when multiplied together producethe given trinomial.Looking for PatternsWhat do you observe in the following Area Models?5

FOA/Algebra 1Unit 3A: Factoring & Solving Quadratic EquationsNotesFactoring using the Area ModelFactor: x2 – 4x – 32STEP 1: ALWAYS check to see if you can factor out a GCF.STEP 2: Multiply the coefficients of the “a” and “c” termstogether and place that number in the bottom ofthe “number diamond” Place the coefficient of the “b” term in the top. Make a factor t-chart for the factors of “a.c” Determine what two numbers can be multiplied toget your “a c” term and added to get your “b”term.a cFactors of a cSum bSTEP 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 your number diamond and place anx after them.STEP 4: Factor out a GCF from each row and column tocreate the binomials or factors you are looking for.STEP 5: Check your factors on the outside by multiplyingthem together to make sure you get all theexpressions in your box.Factor the following trinomials.x2 6x 8Factored Form:Factored Form:6

FOA/Algebra 1Unit 3A: Factoring & Solving Quadratic EquationsNotesPractice Factoring A 1Using the Area Model. Factor the following trinomials.a. Factor x2 4x – 32b. Factor x2 5x 6c. Factor x2 – 3x – 18d. Factor x2 – 14x 48e. Factor x2 – 36f. Factor x2 10x 50Look at these examples. (HINT: Is there a GCF?)g. 2x2 16x 24h. 4x3 12x2 8xRemember your factored form should always been equivalent to the polynomial you started with soyou must always include the GCF on the outside of the factored form.7

FOA/Algebra 1Unit 3A: Factoring & Solving Quadratic EquationsNotesDay 3 – Factor Trinomials with a 1Standard(s):In 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.Factor: 2 x2 3 x 2STEP 1: ALWAYS check to see if you can factor out a GCF.STEP 2: Multiply the coefficients of the “a” and “c” termstogether and place that number in the top of the“number diamond” Place the coefficient of the “b” term in thebottom. Make a factor t-chart for the factors of “a.c” Determine what two numbers can be multiplied toget your “a c” term and added to get your “b”term.a cFactors of a cSum bSTEP 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 your number diamond and place anx after them.STEP 4: Factor out a GCF from each row and column tocreate the binomials or factors you are looking for.STEP 5: Check your factors on the outside by multiplyingthem together to make sure you get all theexpressions in your box.Factored Form:8

FOA/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 5x 3Factored Form:3. 2x 2 17x 30Factored Form:Look at this example. (HINT: Is there a GCF?)4. 6x2 – 40x 24Factored Form:9

FOA/Algebra 1Unit 3A: Factoring & Solving Quadratic EquationsNotesPractice: Take a look at the following trinomials and factor out the GCF, then use the Area Model tofactor.a. 12x2 56x 64b. 25x2 210x - 400c. 10x2 – 72x 72d. 12x2 30x 30e. 8x2 44x 20Remember your factored form should always been equivalent to the polynomial you started with soyou must always include the GCF on the outside of the factored form.10

FOA/Algebra 1Unit 3A: Factoring & Solving Quadratic EquationsNotesDay 4 – Factor Special ProductsStandard(s):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 – 911

FOA/Algebra 1Unit 3A: Factoring & Solving Quadratic EquationsNotesReview: Factor the following expressions:a. x2 8x 16b. x2 – 2x 1c. 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 12

FOA/Algebra 1Unit 3A: Factoring & Solving Quadratic EquationsNotesExploration with Factoring and Quadratic GraphsExploration:a. Given the equation and the graph, what are the zeros in the following graphs?Graph 1: y x2 – 4x 3Graph 2: y 2x2 3x - 2Zeros:Zeros:b. Were you able to accurately determine the zeros?c. What if the equation was written in factored form? Can you accurately name the zeros?Graph 1: y (x – 3)(x – 1)Graph 2: y (x 2)(2x – 1)d. What is the value of y when the parabola crosses the x-axis for each graph?Zero Product Property and Factored FormA polynomial or function is in factored form if it is written as the product of two or more linear binomial factors.Zero Product Property The zero product property is used to solve an equation when one side is zero and the other side is aproduct of binomial factors. The zero product property states that if a· b 0, then a 0 or b 0Examples:a. (x – 2)(x 4) 0b. x(x 4) 0c. (x 3)2 014

FOA/Algebra 1Unit 3A: Factoring & Solving Quadratic EquationsPractice: Identify the zeros of the functions:a. y (x 4)(x 3)b. f(x) (x – 7)(x 5)c. y x(x – 9)Notesd. f(x) 5(x – 4)(x 8)Practice: Create an equation to represent the following graphs:Zeros: x &Zeros: x &y y Solving a quadratic equationreally means:VbThe place(s) where the graph crosses the x-axis has several names. They can be referred as:15

FOA/Algebra 1Unit 3A: Factoring & Solving Quadratic EquationsNotesReview: Methods for FactoringBefore you factor any expression, you must always check for and factor out a Greatest Common Factor (GCF)!A 1GCF (Two Terms)Looks LikeHow to FactorFactor out what is common to both terms(mentally or list method)ax2 - bxExamplesx2 5x x(x 5)18x2 – 6x 6x(3x – 1)-9x2 – x -x(9x 1)Think of what two numbers multiply to get thec term and add to get the b term (Think of thediamond). You also need to think about thesigns:x2 bx cx2 bx c (x #)(x #)x2 – bx c (x - #)(x - #)x2 – bx – c/x2 bx – c (x #)(x - #)x2 8x 7 (x 7)(x 1)x2 – 5x 6 (x – 2)(x – 3)x2 – x – 56 (x 7)(x – 8)Area Model: 3x2 - 5x - 12A not 19x2 – 11x 2 (9x – 2) (x – 1)ax2 bx c2x2 15x 7 (2x 1)(x 7)3x2 – 5x – 28 (2x 7)(x – 4)Difference of Two Squaresx2 – cPerfect SquareTrinomialsFactored Form : (x – 3) (3x 4)x2 bx c“c” is a perfectsquare“b” is double thesquare root of cBoth your “a”and “c” terms should be perfectsquares and since there is no “b” term, it has avalue of 0. You must also be subtracting the aand c terms. Your binomials will be the exactsame except for opposite signs.Difference of Squaresa2 – b2 (a b)(a – b)Factor like you would for when a 1x2 – 9 (x 3)(x – 3)x2 – 100 (x 10)(x – 10)4x2 – 25 (2x 5)(2x – 5)x2 – 6x 9 (x – 3)(x – 3) (x – 3)2x2 16x 64 (x 8)(x 8) (x 8)216

FOA/Algebra 1Unit 3A: Factoring & Solving Quadratic EquationsNotesSolving Quadratic Equations by Factoring & Using the Zero Product PropertyFactor and solve: 2x2 3x 21. Rewrite the equation so it is set equal to 0.2. Check for any GCF’s. Then factor.3. Using the Zero Product Property, set each factorequal to 0.Solve each equation.1: Factoring & Solving Quadratic Equations - GCFSolve the following quadratic equations by factoring (GCF) and using the Zero Product Property.Practice: Solve the following equations by factoring out the GCF.1. 2x2 – 6x 02. x2 x 0Factored Form:Factored Form:Zeros:Zeros:3. -3x2 – 12x 04. 3x2 18xFactored Form:Factored Form:Zeros:Zeros:17

FOA/Algebra 1Unit 3A: Factoring & Solving Quadratic EquationsNotes2: Factoring & Solving Quadratic Equations when a 1Solve the following quadratic equations by factoring and using the Zero Product Property.1. x2 6x 8 02. y x2 – 6x 9Factored Form:Factored Form:Zeros:Zeros:3. x2 4x 324. 5x x2 – 6Factored Form:Factored Form:Zeroes:Zeroes:5. –x2 2x 16. y 𝑥 2 9Factored Form:Factored Form:Zeroes:Zeroes:18

FOA/Algebra 1Unit 3A: Factoring & Solving Quadratic EquationsNotes3: Factoring & Solving Quadratic Equations when a not 1Solve the following quadratic equations by factoring and using the Zero Product Property.1. y 5x2 14x – 3Factored Form:Zeroes:2. 2x2 – 8x – 42 0Factored Form:Zeroes:3. 7x2 – 16x -9Factored Form:Zeroes:4. 6x2 3 11xFactored Form:Zeroes:19

FOA/Algebra 1Unit 3A: Factoring & Solving Quadratic EquationsNotesPutting It All TogetherCan you find the factored AND standard form equations for these graphs?(Remember – standard form is y ax2 bx c)Factored Form:Factored Form:Standard Form:Standard Form:How do you transform an equation from factored form TO standard form?How do you transform an equation from standard form TO factored form?20

FOA/Algebra 1Unit 3A: Factoring & Solving Quadratic EquationsNotesDay 7 – Solving by Finding Square RootsStandard(s):Review: If possible, simplify the following radicals completely.a.b.25c.12524Explore: Solve the following equations for x:a. x2 16b. x2 4c. x2 9d. x2 1What operation did you perform to solve for x?How many of you only had one number as an answer for each equation?Well, let’s take a look at the graph of this function.After looking at the graph, what values of x produce a y value of 1, 4, 9, and 16?What would be your new answers for the previous equations?a. x2 16b. x2 4c. x2 9d. x2 121

FOA/Algebra 1Unit 3A: Factoring & Solving Quadratic EquationsNotesSolving by Taking Square Roots without ParenthesesSteps for Solving Quadratics by Finding Square Roots1. Add or Subtract any constants that are on the same side of x 2.2. Multiply or Divide any constants from x2 terms. “Get x2 by itself”3. Take square root of both sides and set equal to positive and negative roots ( ).Ex: x2 25 x2 25x 5x 5 and x - 5REMEMBER WHEN SOLVING FOR X YOU GET A AND ANSWER!Solve the following for x:1) x 2 492) x 2 203) x 2 76) x 2 11 144)3x 2 1085) 2 x 2 1287)7 x 2 6 578)10) 10 x 2 9 4992x 2 8 17011) 4 x 2 6 749)x2 012) 3 x 2 7 30122

FOA/Algebra 1Unit 3A: Factoring & Solving Quadratic EquationsNotesSolving by Finding Square Roots (More Complicated)Steps for Solving Quadratics by Finding Square Roots with Parentheses1. Add or Subtract any constants outside of any parenthesis.2. Multiply or Divide any constants around parenthesis/squared term. “Get ( ) 2 by itself”3. Take square root of both sides and set your expression equal to BOTH the positive andnegative root ( ). Ex: (x 4)2 252 (x 4) 25(x 4) 5x 4 5 and x 4 -5x 1 and x – 94. Add, subtract, multiply, or divide any remaining numbers to isolate x.REMEMBER WHEN SOLVING FOR X YOU GET A POSITIVE AND NEGATIVE ANSWER!Solve the following for x:1)( x 4)2 812)( p 4)2 163)10( x 7)2 4404)12 x 8 1425) 2( x 3)2 16 486)3( x 4)2 7 6723

FOA/Algebra 1Unit 3A: Factoring & Solving Quadratic EquationsNotesDay 8 – Solving by Completing the SquareStandard(s):Some trinomials form special patterns that can easily allow you to factor the quadratic equation. We will lookat two special cases:Review: Factor the following trinomials.1. x2 – 6x 92. x2 10x 253. x2 – 16x 64(a) How does the constant term in the binomial relate to the b term in the trinomial?(b) How does the constant term in the binomial relate to the c term in the trinomial?Problems 1-3 are called Perfect Square Trinomials. These trinomials are called perfect squaretrinomials because when they are in their factored form, they are a binomial squared.An example would be x2 12x 36. Its factored form is (x 6)2, which is a binomial squared.But what if you were not given the c term of a trinomial? How could we find it?Complete the square to form a perfect square trinomial and then factor.a. 𝑥 2 12𝑥 b. 𝑧 2 4𝑧 c. 𝑥 2 18𝑥 24

FOA/Algebra 1Unit 3A: Factoring & Solving Quadratic EquationsNotesSolving equations by “COMPLETING THE SQUARE”The Equation:STEP 1: Write the equation in the formx2 bx c (Bring the constant to the other side)STEP 2: Make the left hand side a perfect squarex2 6 x 2 0x 2 6 x 2 𝑥 2 6𝑥 (3)2 2 (3)22 b trinomial by adding to both sides 2 STEP 3: Factor the left side, simplify the right side( x 3)2 7STEP 4: Solve by taking square roots on both sidesx 3 7 and x 3 7x 7 - 3 and x - 7 - 3Group Practice: Solve for x by “Completing the Square”.1. x2 – 6x - 72 02. x2 80 18xX X 3. x2 – 14x – 59 -204. 2x2 - 36x 10 0X X 25

FOA/Algebra 1Unit 3A: Factoring & Solving Quadratic EquationsNotesThe DiscriminantInstead of observing a quadratic function’s graph and/or solving it by factoring, there is an alternative way todetermine the number of real solutions called the discriminant.Interpretation of the Discriminant (b2 – 4ac)Given a quadratic function in standard form:ax2 bx c 0, where a 0 ,The discriminant is found by using: b2 – 4acThis value is used to determine the number of realsolutions/zeros/roots/x-intercepts that exist for a quadraticequation. If b2 – 4ac is positive: If b2 – 4ac is zero: If b2 – 4ac is negative:Practice: Find the discriminant for the previous three functions:a.)f ( x) x 2 4 x 3a b c Discriminant:#. of real solutions:b.)f ( x) x 2 10x 25a b c Discriminant:# of real zeros:c.)f ( x) x 2 x 1a b c Discriminant:# of real roots:Practice: Determine whether the discriminant would be greater than, less than, or equal to zero.27

FOA/Algebra 1Unit 3A: Factoring & Solving Quadratic Equations3) 7x2 8x 3 0 a b c 4) 3x2 2x 8 a b c Discriminant:Discriminant:X Roots:Approx:Approx:Notes29

FOA/Algebra 1Unit 3A: Factoring & Solving Quadratic EquationsNotesDay 10 – Applications of QuadraticsStandard(s):If you are solving for the vertex:-Maximum/Minimum (height, cost, etc)-Greatest/Least Value-Maximize/Minimize-Highest/LowestFalling Objects:Thrown Object:h -16t2 h0h -16t2 vt h0If you are solving for the zeros:-How long did it take to reach the ground?-How long is an object in the air?-How wide is an object?-Finding a specific measurement/dimensionVertical Motion Modelsh0 starting height, h ending heighth0 starting height, h ending height,v velocity (going up ( ) or down (-))30

FOA/Algebra 1Unit 3A: Factoring & Solving Quadratic EquationsNotesScenario 1. Suppose the flight of a launched bottle rocket can be modeled by the equation y -x2 6x, wherey measures the rocket’s height above the ground in meters and x represents the rocket’s horizontal distance inmeters from the launching spot at x 0.a. How far has the bottle rocket traveled horizontally when it reaches it maximum height? What is themaximum height the bottle rocket reaches?b. When is the bottle rocket on the ground? How far does the bottle rocket travel in the horizontal directionfrom launch to landing?Scenario 2. A frog is about to hop from the bank of a creek. The path of the jump can be modeled by theequation h(x) -x2 4x 1, where h(x) is the frog’s height above the water and x is the number of secondssince the frog jumped. A fly is cruising at a height of 5 feet above the water. Is it possible for the frog to catchthe fly, given the equation of the frog’s jump?b. When does the frog land back in the water?31

FOA/Algebra 1Unit 3A: Factoring & Solving Quadratic EquationsNotesScenario 3. A school is planning to host a dance with all profits going to charity. The amount of profit is foundby subtracting the total costs from the total income. The income from ticket sales can be expressed as 200x –10x2, where x is the cost of a ticket. The costs of putting on the dance can be expressed as 500 20x.a. What are the ticket prices that will result in the dance breaking even?b. What are ticket prices that will result in a profit of 200?Applications of Solving by Square RootsFalling Objects:h -16t2 h0h0 starting height, h ending heightScenario 4. The tallest building in the USA is in Chicago, Illinois. It is 1450 ft tall. How long would it take a pennyto drop from the top of the building to the ground?Scenario 5. When an object is dropped from a height of 72 feet, how long does it take the object to hit theground?32

FOA/Algebra 1Unit 3A: Factoring & Solving Quadratic EquationsNotesDay 11 – Determining the Best MethodStandard(s):Completing the Squareax2 bx c 0,when a 1 and b is an even #Non Factorable MethodsFinding Square Rootsax2 - c 0Parenthesis in equationExamplesx2 – 6x 11 0x2 – 2x - 20 0Examples2x2 5 95(x 3)2 – 5 20x2 – 36 0Factorable MethodsA 1 & A Not 1 (Factor into 2 Binomials)ax2 bx c 0, when a 1ax2 bx 02ax bx c 0, when a 1x2 - c 0Examples5x2 20x 0Examplesx2 – 6x 8x3x2 – 20x – 7 0x2 – 3x 2 0x2 5x -6x2 – 25 0Quadratic Formulaax2 bx c 0Any equation in standard formLarge coefficientsExamples3x2 9x – 1 020x2 36x – 17 0GCFDetermine the best method for solving. Explain why.1.6x 2 11x 3 03. x2 – 7x 82. x2 6x – 45 04. 8x2 24x 033

FOA/Algebra 15. 2x2 – 11x 5 07.x2 9 0Unit 3A: Factoring & Solving Quadratic EquationsNotes6. x2 – 9x -208. x2 4x 17 09. 2x2 6x – 37 010. 4(x 4)2 1611. x2 – 15x 36 012. 18x2 100x 6334

Factor special products (difference of two squares) Learning 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 of a function. Learning Tar