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Algebra 2: Unit 5 Continued FACTORING QUADRATIC EXPRESSION Factors Factors are numbers or expressions that you multiply to get another number or expression. Ex. 3 and 4 are factors of 12 because 3x4 = 12 Factors What are the following expressions factors of? 1. 4 and 5? 2. 5 and (x + 10) 3. 4 and (2x + 3) 4. (x + 3) and (x - 4) GCF One way to factor an expression is to factor out a GCF or a GREATEST COMMON FACTOR. EX: 4x2 + 20x – 12 EX: 9n2 – 24n Try Some! Factor: a. 9x2 +3x – 18 b. 7p2 + 21 c. 4w2 + 2w Factors of Quadratic Expressions When you multiply 2 binomials: (x + a)(x + b) = x2 + (a +b)x + (ab) This only works when the coefficient for x2 is 1. Finding Factors of Quadratic Expressions When a = 1: x2 + bx + c Step 1. Determine the signs of the factors Step 2. Find 2 numbers that’s product is c, and who’s sum is b. Sign table! 2nd sign + Same Different Question 1st sign Answer + - + or - (x+ )(x+ ) (x )(x ) (x + )(x - ) OR (x - )(x + ) Examples Factor: 1. x2 + 5x + 6 2. x2 – 10x + 25 3. x2 – 6x – 16 4. x2 + 4x – 45 Examples Factor: 1. x2 + 6x + 9 2. x2 – 13x + 42 3. x2 – 5x – 66 4. x2 – 16 Box and Slide Method When a does NOT equal 1. Steps 1. Slide 2. Factor 3. Divide 4. Reduce 5. Slide Example! Factor: 1. 3x2 – 16x + 5 Example! Factor: 2. 2x2 + 11x + 12 Example! Factor: 3. 2x2 + 7x – 9 You Try! Factor 1. 5t2 + 28t + 32 2. 2m2 – 11m + 15 Quadratic Equations 5 ways to solve There are 5 ways to solve quadratic equations: 1. Factoring 2. Finding the Square Root 3. Graphing 4. Completing the Square 5. Quadratic Formula Quadratic Equation Standard Form of Quadratic Function: y = ax2 + bx + c Standard Form of Quadratic Equation: 0 = ax2 + bx + c Solutions A SOLUTION to a quadratic equation is a value for x, that will make 0 = ax2 + bx + c true. Note: A quadratic equation always have 2 solutions. SOLVING BY FACTORING Factoring Solve by factoring; 2x2 – 11x = -15 Factoring Solve by factoring; x2 + 7x = 18 Solve by Factoring: 1. 2x2 + 4x = 6 2. 16x2 – 8x = 0 Solving by Finding Square Roots For any real number x; x2 = n x=± n Example: x2 = 25 Solve Solve by finding the square root; 5x2 – 180 = 0 Solve Solve by finding the square root; 4x2 – 25 = 0 Try Some! Solve by finding the Square Root: 1. x2 – 25 = 0 2. x2 – 15 = 34 3. x2 – 14 = -10 4. (x – 4)2 = 25 Quadratic Equations SOLVING BY GRAPHING Solving by Graphing For a quadratic function, y = ax2 +bx + c a zero of the function (where a function crosses the x-axis) is a solution of the equations ax2 + bx + c = 0 Examples Solve x2 – 5x + 2 = 0 Examples Solve x2 + 6x + 4 = 0 Examples Solve 3x2 + 5x – 12 = 8 Examples Solve x2 = -2x + 7 Complex Numbers Simplifying Radicals If the number has a perfect square factor, you can bring out the perfect square. EX: 18 75 48 You Try! 27 200 75 Try this: Solve the following quadratic equations by finding the square root: 4x2 + 100 = 0 What happens? Complex Numbers Imaginary Number: i The Imaginary number i = -1 This can be used to find the root of any negative number. r i r For example: -9 -75 Properties of i i = -1 i = 2 ( -1) 2 = -1 i = i (i) = -1i 3 2 i = i (i ) = (-1)(-1) =1 4 2 2 Graphing Complex Number Absolute Values Absolute Values Operations with Complex Numbers The Imaginary unit, i, can be treated as a variable! Adding Complex Numbers: (8 + 3i) + ( -6 + 2i) You Try! 1. 7 – (3 + 2i) 2. (4 – 6i) + 3i Operations with Complex Numbers Multiplying Complex Numbers: Example: (5i)(-4i) Example: (2 + 3i)(-3 + 5i) Try Some! 1. (6 – 5i)(4 – 3i) 2. (4 – 9i)(4 + 3i) Now we can SOLVE THIS! Solve 4x2 + 100 = 0 Completing the Square 5 ways to solve There are 5 ways to solve quadratic equations: Factoring Finding the Square Root Graphing Completing the Square Quadratic Formula Solving a Perfect Square Trinomial We can solve a Perfect Square Trinomial using square roots. A Perfect Square Trinomial is one with two of the same factors! X2 + 10x + 25 = 36 Solving a Perfect Square Trinomial x2 – 14x + 49 = 81 What if it’s not a Perfect Square Trinomials?! If an equation is NOT a perfect square Trinomial, we can use a method called COMPLETING THE SQUARE. Completing the Square Using the formula for completing the square, turn each trinomial into a perfect square trinomial. Solving by Completing the Square Solve by completing the square: x2 + 6x + 8 = 0 Solving by Completing the Square Solve by completing the square: x2 – 12x + 5 = 0 Solving by Completing the Square Solve by completing the square: x2 – 8x + 36 = 0 Classwork: Solve by completing the square 1) x 4 x 21 2) x 8x 33 3) x 10x 5 4) x 5x 5 0 5) x x70 2 2 2 2 2 Solving Quadratic Equations Solve by Factoring 2x2 – x = 3 x2 + 6x + 8 = 0 Solve by Finding the Square Root 5x2 = 80 2x2 + 32 = 0 Solve by Graphing X2 + 5x + 3 = 0 3x2 – 5x – 4 = 0 Solve by Completing the Square X2 – 3x = 28 x2 + 6x – 41 = 0 5 ways to solve There are 5 ways to solve quadratic equations: Factoring Finding the Square Root Graphing Completing the Square Quadratic Formula Quadratic Formula The Quadratic Formulas is our final way to Solve! It works when all else fails! Examples 2x2 + 6x + 1 = 0 Examples X2 – 4x + 3 = 0 3x2 + 2x – 1 = 0 X2 = 3x – 1 8x2 – 2x – 3 = 0 Discriminant Discriminant Discriminant 1. IF the Discriminant is POSITIVE then there are 2 REAL solutions 2. IF the Discriminant is ZERO then there is ONE REAL solution 3. IF the Discriminant is NEGATIVE then there are 2 IMAGINARY solutions. Using the Discriminant The weekly revenue for a company is: R = -3p2 + 60p + 1060, where p is the price of the company’s product. Use the discriminant to find whether there is a price the company can sell their product to reach a maximum revenue of $1500?