Solving Quadratic Equations by FACTORING

What are Quadratic Equations? A quadratic equation is an equation which: y = x2 y = x2 + 2 y = x2 + x – 4 y = x2 + 2x – 3 Contains a x2 term All of these equations contain a x2 term therefore they are called: Quadratic Equations

Which of the following are Quadratic Equations? y = x + 3 It Contains a x2 term y = x2 ....WHY? y = 2x – 4 It Contains a x2 term y = x2 + 2x – 3 ….WHY?

Solving Quadratic Equations BY FACTORING Remember: Quadratic Equations Contain a x2 term There are several methods of solving QUADRATICS but one methods that you must know is called FACTORING “Factors” are the numbers you multiply to get another number 1 x 6 and 2 x 3 The (+) factors of 6 are: The (-) factors of 6 are: -1 x -6 and -2 x -3

BIG IDEA NUMBER ONE Solving Quadratic Equations BY FACTORING If A(B) = 0 what can we say about either A or B? Either A or B must equal ZERO!!! A = 0 or B = 0

Solving Quadratic Equations BY FACTORING BIG IDEA NUMBER ONE So if… (x + 3) (x – 3) = 0 THEN EITHER (x + 3) = 0 or (x – 3) = 0 So…. x = -3 or x = 3

BIG IDEA NUMBER ONE Solving Quadratic Equations BY FACTORING TO SOLVE A QUADRATIC EQUATION BY FACTORING MAKE THE EQUATION EQUAL TO ZERO FACTOR THE EQUATION SET THE FACTORS EQUAL TO ZERO AND SOLVE

How to solve Quadratic Equations by FACTORING Example 1 x2 + x + = 0 7 7 12 12 1 x 12 =12 -1 x -12 = 12 2 x 6 = 12 -2 x -6 = 12 3 x 4 = 12 -3 x -4 = 12 Write down all the factor pairs of ___. (x )(x ) = 0 What goes with the x? 1 Positive Negative From this list, choose the pair that adds up to ___ 2 3 + 4 = 7 0 = (x + 3)(x + 4) x = – 3 and – 4 0 = (x + )(x + ) 0 = (x + 3)(x + 4) 3 Put these numbers into brackets

(x + 3) (x + 4) x(x + 4) + 3(x + 4) x(x) + x(4) + 3(x) + 3(4) PROOF: x2 + 7x + 12 = (x + 3) (x + 4) (x + 3) (x + 4) x(x + 4) + 3(x + 4) x(x) + x(4) + 3(x) + 3(4) x2 + 4x + 3x + 12 x2 + 7x + 12 Factor: Factor: Combine like terms:

How to solve Quadratic Equations by FACTORING Example 2 x2 x + = 0 - 5 - 5 6 6 1 x 6 = 6 -1 x -6 = 6 2 x 3 = 6 -2 x -3 = 6 Write down all the factor pairs of ___ . 1 Positive Negative From this list, choose the pair that adds up to ___ . 2 -2 + -3 = -5 (x - 2)(x - 3) = O x = 2 and 3 3 Put these numbers into brackets

Solve by factoring: x2 + x - 6 = 0 Write down all the factor pairs of – 6 1 x -6 = -6 2 x -3 = -6 3 x -2 = -6 6 x -1 = -6 1 From this list, choose the pair that adds up to 1 2 (3) + (-2) = 1 3 – 2 = 1 0 = (x + 3)(x - 2) x = – 3 and 2 Put these numbers into brackets 3

PLEASE TAKE OUT YOUR QUADRATIC EQUATIONS POWERPOINT USE WORKSHEET #1 x2 + 3x + 2 = 0 Find all the factor pairs of _____ From these choose the pair that add up to _____ Put these values into the brackets (x _)(x _) = 0 x2 + x – 12 = 0 From these choose the pair that add up to ­_____ (x + _)(x + _) = 0 x2 – 12x – 20 = 0 . Copy and fill in the missing values when you factor x2 + 8x + 12 = 0 Find all the factor pairs of _____ From these choose the pair that add up to _____ Put these values into the brackets (x _ )(x _ ) = 0 x = -2 x = -6 2 1 x 2 = 2 -1 x -2 = 2 PLEASE TAKE OUT YOUR QUADRATIC EQUATIONS POWERPOINT WORKSHEET # 1 3 1 + 2= 3 + 1 + 2 12 1 x 12 =12 -1 x -12 = 12 2 x 6 = 12 -2 x -6 = 12 3 x 4 = 12 -3 x -4 = 12 WORK TOGETHER TO FACTOR THE NEXT QUADRATIC 8 2 + 6 = 8 +2 +6

PLEASE TAKE OUT YOUR QUADRATIC EQUATIONS POWERPOINT 1 x2 + 5x + 6 = 0 2 x2 - x – 6 = 0 3 x2 + 8x + 12 = 0 4 x2 + x – 12 = 0 5 x2 - 8x + 15 = 0 6 x2 + 3x – 28 = 0 7 x2 - 3x – 18 = 0 8 x2 - 10x – 24 = 0 9 x2 + 8x + 16 = 0 10 x2 - 6x – 40 = 0 (x + 3)(x + 2) (x – 3)(x + 2) (x + 2)(x + 6) (x – 3)(x + 4) (x – 3)(x – 5) (x + 7)(x – 4) PLEASE TAKE OUT YOUR QUADRATIC EQUATIONS POWERPOINT WORKSHEET # 2 (x – 6)(x + 3) (x - 12)(x + 2) (x + 4)(x + 4) (x - 10)(x + 4)

1 x2 + 5x + 6 = 0 (x + 3)(x + 2) 2 x2 - x – 6 = 0 (x – 3)(x + 2) 3 4 x2 + x – 12 = 0 (x – 3)(x + 4) 5 x2 - 8x + 15 = 0 (x – 3)(x – 5) 6 x2 + 3x – 21 = 0 (x + 7)(x – 4) 7 x2 - 3x – 18 = 0 (x – 6)(x + 3) 8 x2 - 10x – 24 = 0 (x - 12)(x + 2) 9 x2 + 8x + 16 = 0 (x + 4)(x + 4) 10 x2 - 4x – 60 = 0 (x - 10)(x + 4) -3 and -2 3 and -2 -2 and -6 3 and -4 3 and 5 -7 and 4 6 and -3 6 and -3 -4 and -4 - 10 and -4

FACTORING SPECIAL QUADRATIC EQUATIONS THE DIFFERENCE BETWEEN PERFECT SQUARES

FACTORING THE DIFFERENCE BETWEEN PERFECT SQUARES (x2 + 0x - 4) Is This A Quadratic Equation? FACTORING (x2 + 0x - 4) 1 x -4 = -4 2 x -2 = -4 1 Find all the factor pairs of - 4 2 From these choose the pair that add up to “0” 2 + -2 = 0 3 Put these values into the brackets (x + _)(x + _) = 0 (x + 2)(x - 2) = 0 Notice: x2 + 0x – 4 = (x2 – 4)

FACTORING THE DIFFERENCE BETWEEN PERFECT SQUARES This is often called the “Difference between Two Squares” x2 – 4 (x + 2)(x – 2)

FACTORING THE DIFFERENCE BETWEEN PERFECT SQUARES TO FACTOR THE DIFFERENCE BETWEEN SQUARES x2 - 16 1) TAKE THE SQUARE ROOT OF THE BOTH TERMS . x2 = x 16 = 4 2) MAKE THE BRACKETS { one (+) one (-) } AND FILL IN THE BLANKS. (x + __ ) (x - __ ) 4 4 x2 – 16 = (x + 4 ) (x - 4 ) (x + 4 ) = 0 (x - 4 ) = 0 x = -4 x = 4

To Show Geometrically That (a + b)2 = a2 + 2ab + b2 a + b Now.. Cross Multiply a a2 ab a2 +ab + b ab b2 + b2 +ab a2 + 2ab + b2

To Show Algebraically That (a + b)2 = a2 + 2ab + b2 (a + b) (a + b) a2 + 2ab + b2 a b (a + b) (a + b) + a(a) ab + +

This is often called the difference between two squares -1 x 4 = -4 -2 x 2 = -4 4 x -1 = -4 -2 + 2 = 0 x2 – 4 x2 + 0x – 4 (x – 2)(x + 2) Notice that x2 – 4 could be written as x2 – 22 (x – 2)(x + 2) This is often called the difference between two squares x2 – 25 (x + 5)(x – 5)

USE YOUR WORKSHEET TO SOLVE THE DIFFERENCE OF SQUARES 1) MAKE THE BRACKETS { one (+) one (-) } 2) TAKE THE SQUARE ROOT OF THE NUMBER AND FILL IN THE BLANKS 1 x2 - 9 2 x2 - 100 3 x2 - 36 4 x2 - 49 5 x2 - 81 (x + 3) = 0 (x – 3) = 0 x = -3 x = 3 x = 3 or -3 (x + __ ) (x - __ ) 3 3

1 x2 - 9 (x + 3)(x – 3) 2 x2 - 100 (x + 10)(x – 10) 3 x2 - 36 (x + 6)(x – 6) 4 x2 - 49 (x + 7)(x – 7) 5 x2 - 81 (x + 9)(x – 9) 6 x2 - 64 (x + 8)(x – 8) 7 x2 - 18 (x + √18)(x – √18) 8 x2 - 24 (x + √24)(x – √24)

(x )(x ) What goes with the x?

(x + 3)(x + 2) x(x + 2) + 3(x + 2)  x X (x + 2) + 3 X (x + 2) You try (x + 5)(x + 2) (x – 2)(x + 3) (x + 2)(x – 4) (x – 3)(x – 2) x(x + 2) + 3(x + 2)  x X (x + 2) + 3 X (x + 2)  x X x + x X 2 + 3 X x + 3 X 2  x2 + 2x + 3x + 6  x2 + 5x + 6

(x - 3) (x - 2) x(x - 2) -3(x - 2) x(x) + x(-2) - 3(x) - 3(-2) PROOF: x2 - 5x + 6 = (x - 3) (x - 2) (x - 3) (x - 2) x(x - 2) -3(x - 2) x(x) + x(-2) - 3(x) - 3(-2) x2 - 2x - 3x + 6 x2 - 5x + 6 Factor: Factor: Combine like terms: