CP Algebra 2 Unit 2-1 Factoring And Solving Quadratics

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Unit 2-1Factoring and Solving QuadraticsLearning Targets:1. I can factor using GCF.2. I can factor by grouping.FactoringQuadraticExpressions3. 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.7. I can solve by factoring.8. I can solve by taking the square root.9. I can perform operations with imaginary numbers.SolvingQuadraticEquations10. I can solve by completing the square.11. I can solve equations using the quadratic formula (with rationalized denominators).12. I can use the discriminant to determine the number and type of solutions.13. I can write quadratic equations given the real solutions.2

Factoring BasicsDate:After this lesson and practice, I will be able to ! factor using GCF. (LT 1)factor by grouping. (LT 2)! factor when a is one. (LT 3)factor when a is not one. (LT hen we’ve modeled real-life situations with functions before, it’s been withfunctions (example: at-bats vs. runs in baseball, GB vs. cost of a data plan, etc.).Suppose we wanted to determine how high you could throw a tennis ball? What would the path of theball look like?As you can see, in order to model situations such as these, we require a function family that is not. For situations such as these, we can use the family offunctions.(Throughout the year, we will be studying several different families of functions. We will learn how tograph and solve each type of function. We’re going to start Quadratics with solving!)Quadratic Function: A function that can be written in the formwhere .Standard Form of a Quadratic Function:You might recognize these from previous math classes! Try to solve a quadratic using the usual methodwe used to solve linear equations 𝑥 ! 8𝑥 7 5It turns out we’ll need some new (and some not so new) skills in order to solve quadratic equations.One of those essential skills is factoring.3

To factor completely is to write a polynomial as a product of polynomialswith integer coefficient. Make sure your final answers contain only unfactorable pieces.Factor Using GCF (LT 1)This is the strategy you should use when factoring any expression.Example 1: Factor completely.A) 14 x 2 21xB) 80n7 8n 4 16n3C) 2x 3 y 2 yWhile the following technique does not necessarily apply exclusively to expressions,it will provide us the skills necessary to factor quadratics.Factor By Grouping (LT 2)Factor by grouping - pairing together binomials that have common monomial factors:Example 2: Factor completely.A) 5x 3 35x 2 2x 14B) 3n3 18n2 2n 12Example 3: Factor completely.A) x3 6 x 2 7 x 42B) 9r 3 15r 2 3r 5Factoring when a is one (LT 3)You probably remember doing this kind of factoring in Algebra 1 and/or Geometry.Example 4: Factor completely.A) x 2 11x 10B) x 2 4 x 21C) x3 2 x 2 15 x4

Example 5: Factor completely.A) x 2 17 x 72B) x 2 8 x 12C) 4 x 2 20 x 24D) 6 x5 30 x 4 36 x3Factoring when a is not one (LT 4)Example 6: Factor completely.A) 7 x 2 43x 6B) 12 x 2 37 x 11Example 7: Factor completely.A) 3x 2 11x 8B) 7 x 2 37 x 30C) 18 x 2 90 x 100D) 6 x 2 21x 275

More FactoringDate:After this lesson and practice, I will be able to ! factor perfect square trinomials. (LT 5)! factor using difference of squares. (LT hen factoring, there are types of polynomials that are so common that it is helpful to memorizefor how to factor them quickly.Factoring Perfect Square Trinomials (LT 5)Let’s start by factoring x 2 14 x 49 using our familiar methods:Observe:- The last term is a . The first term is a .- The middle term is the product of the quantities being squared in the andterms.- Algebraically:Example 1: Determine if the quadratic is a perfect square trinomial.A) x 2 50 x 100B) x 2 6 x 9C) 16 x 2 24 x 9Example 2: Factor completely the perfect square trinomials from above.Example 3: Factor completely using the perfect square trinomial pattern.A) 9 x 2 24 x 16C) 25r 2 30r 96

Factoring Using Difference of Two Squares (LT 6)Let’s start by factoring x 2 25 using our familiar methods:Observe:- The last term is a .The first term is a .- The two factors are identical except for the .- Algebraically:Example 4: Factor completely.A) x 2 121B) 4 x 2 49C) x 2 4Example 5: Factor completely.A) 16 x 2 9B) 27m2 75!C) 64𝑥 817

Solving Quadratic EquationsDate:After this lesson and practice, I will be able to ! solve by factoring. (LT 7)! solve by taking the square root. (LT t’s finally time to start solving! The first way of solving combines factoring and the zero productproperty. Here’s what that property states:If ab 0, then .You’ll first want to get your standard form equation set equal to . Then get it in factored form.To get an equation in factored form, just write it in standard form, then factor! Once the quadratic equation is in factored form, use the zero product property. Essentially to do this,you just try to find the value for x that makes the equation .Example 1: Solve by factoring.A)( x 4)(3x 2) 0B)x 2 x 12 0C)2n2 13n 23 3D)15x 6 12x 5 105x 4 84x 3 0E)2 x 2 12 x 80F)7r 2 20r 36 48

Sometimes you won’t be able to factor as nicely as you did above. What’s different about the equationsin Example 2?For these, you can solve by getting x2 alone, and then take the .Example 2: Solve. If necessary, simplify all radicals and give the radical rounded to 2 decimal places.A)5x 2 180 0B)3x 2 24C)4x 2 25 0D)x2 E)(𝑥 4)! 91 04Always make sure your radical answers are always simplified! Here’s a quick review A)180B)252C)54 p3D)481E)7169

Imaginary NumbersDate:After this lesson and practice, I will be able to ! perform operations with imaginary numbers. (LT arm Up: Solve the equation 𝑥 ! 121 0This quadratic equation has real solutions. However, there do exist solutions outside thenumber system. Let’s review that number system An imaginary number is any number whose square is -1. We use the letter 𝒊 to represent imaginarynumbers.Here’s the most important thing: 𝑖 ! 1That means that 𝑖 Here’s a quick example: 4 Example 1: Simplify each radical completely.A) 5B)! 36C)! 12D) 49E)5 54F)! 12010

So that’s imaginary numbers. When you combine a real number with an imaginary number, you get acomplex number. A complex number is a real number plus an imaginary number. The standard formof complex numbers looks like this:Example 2: Write each expression as a complex number in standard form.A) (12 11i ) ( 8 3i)B) (15 9i) (24 9i)Example 3: Write each expression as a complex number in standard form.A) 4i ( 6 i )B) (9 2i)( 4 7i)C) (4 i)(4 i)Example 4: Write each expression as a complex number in standard form.A) (9 i) ( 6 7i)B) 4 (1 i) (2 3i)C) (2 7i)(2 7i)D) 5i (8 9i )F) (3 4i) 2E) ( 8 2i)(4 7i)Let’s return to the equation from the beginning of class: !x 2 121 0 . You are now prepared to solveequations to find all solutions, whether they are or .Example 5: Solve !4x 2 72 011

Completing the SquareDate:After this lesson and practice, I will be able to ! solve by completing the square. (LT Do you remember perfect square trinomials? We learned how to factor them earlier in the unit. Let’suse that to help us solve an equation!Example 1: Solve by finding square roots x 2 8 x 16 25That’s a pretty great way to solve quadratic equations, but unfortunately, we won’t always have niceperfect square trinomials in our equations. Luckily for us, there’s a way to create perfect squaretrinomials. It’s called completing the square In your group, set up the following scenarios with your papers and fill in the table accordingly.EquationNumber of 1-tiles needed toExpression written as a squarecomplete the square𝑥 ! 2𝑥 𝑥 ! 4𝑥 𝑥 ! 6𝑥 𝑥 ! 8𝑥 𝑥 ! 10𝑥 Answer these questions with your group. Consider the general statement 𝑥 ! 𝑏𝑥 𝑐 (𝑥 𝑑)! .a. How is d related to b?b. How is c related to d?c. How can you obtain the numbers in the second column of the table directly from the coefficientsof x in the expressions from the first column?12

Let’s practice completing the square before we solve Example 2: Find the value of c that makes x 2 bx a perfect square trinomial. Then write the expressionas the square of a binomial.A) x 2 16 x cD) x 2 6 x cB) x 2 22 x cC) x 2 9 x cE) x 2 15 x cSolving quadratic equations of the form x 2 bx c 0 by completing the squareExample 3: Solve by completing the square:A) x 2 12 x 4 0B) x 2 8x 36 0Example 4: Solve each equation by completing the square.1.2!x 10x 8 02.2!x 4x 4 013

3.2!x 4x 1 04.2!x 6x 12 0Solving quadratic equations of the form ax 2 bx c 0, a 1 by completing the square.Example 5: Solve by completing the square:A) 2 x 2 8 x 14 0B) 3x 2 36 x 150 0C) 2x 2 x 614

The Quadratic FormulaDate:After this lesson and practice, I will be able to ! solve using the quadratic formula (with rationalized denominators). (LT 11)! use the discriminant to determine the number and type of solutions. (LT 12)! write quadratic questions given the real solutions. (LT So far, you have learned three strategies for solving quadratic equations they are:Each strategy has its own benefit depending on the characteristics of the equation. Today you’re goingto see the fourth and final way of solving quadratic equations. It’s one of the most famous formulas inmathematics and it works on quadratic equation!The Quadratic Formula: For any quadratic equation, the solutions are .Example 1: Solve each equation. Write your solutions as exact values in simplest form (no decimals)A) 3x 2 5x 2B) !2x 2 8x 12C) 4 x 2 8 x 1 0D) !2x 2 7x 8E) 9x 2 12x 4 0As we’ve seen, quadratic equations can have or solutions. Let’sdiscover how to quickly determine which type of solution a given quadratic equation has Discriminant: Given ax 2 bx c 0 , the discriminant is .15

Fill out the table below with your group (split up the work if you want).EquationValue of DiscriminantNumber and Type ofSolutions2!8x 2x 3 02!2x 8x 12 02!x 10x 25 0The discriminants of different quadratic equations are given below. Look for patterns in the table above,then predict how many real solutions each quadratic equation has.DiscriminantNumber of Real Solutions-5160Summary:Value ofDiscriminantNumber and Types of SolutionsPositiveZeroNegativeExample 2: Find the discriminant and give the number and type of solutions. Do not solve.A) x 2 10 x 23 0B) x 2 10 x 25 0C) x 2 10 x 27 016

We now know how to find the solutions of a quadratic equation. Next we will do the reverse: find thequadratic equation given the solutions. We will use this quite a bit in the next unit.Factored Form – , given r1 and r2 are zeros.!!Example 1: Write a quadratic equation in factored form with the following solutions:1 and -7A)2 and -4B)4C)! 2Example 2: Write a quadratic equation in factored form with the following solutions.A)-1 and 3B)-4 and 0Example 3: Write a quadratic equation in STANDARD form with the following solutions.A)-2 and 2B)-5 and 117


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.

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