Amani Mubarak 9-5

2X²-8X+6 1.First multiply aXc. 2.Now find two factors of that multiply to the answer of aXc, and that will also Add up to b. *USE A T-TABLE TO MAKE IT EASIER. (In one side of t-table you write b and in the other side the answer of aXc.) 1. 2X6=12-2 X X= -1, -3

3x²-2x-16= X X= 8/3, -2 x² +7x +15= x5 2+5 x² + 7x + 15= 0 x x= 2,5 x² + 8x = X²+8x+15= X53+5 x x= 3,5

QUADRATIC FUNCTION  A Quadratic Function is written in the form: f(x) = ax 2 + bx + c.  The graph of a quadratic function is a curve called a parabola. LINEAR FUNCTION  A Linear Function is written in the form: y= mx + b  The graph of a linear function is a straight line.

 Its very simple to tell the difference between this two types of functions, since a graph of a quadratic function will have a curved line, parabola, and a linear function is just a straight line. y= 1/2x + 2 y=3x² + 6x + 1

y= -x + 5 y= 1/2x²

HOW TO GRAPH A QUADRATIC FUNCTION  f (x) = ax 2 + bx + c  Y intercept of the graph is found by f(0)=c  X intercept of the graph is found by solving the equation: ax 2 + bx + c = 0  ax 2 + bx + c = 0 is solved by using –b/2a  STEPS: 1. set = 0 2. graph the function a. make a t-table b. find the vertex x= -b÷2ª c. pick 2 points to the left and 2 to the right. d. graph the parable 3. find x-values where it crosses the x-axis.

Examples: 1. y= -x² b/2ª= 0 xy X=0,0

2. y= 1/2x² b/2(a)=0 xy X=0,0

3. y=3x² + 6x + 1 -b/2(a)= -6/2 (3) -6/6= -1 XY X= -1,-2

4. y= x² + 2x + 5 -b/2(a)= -2/2(0)= -2/2 xy

How to solve a quadratic equation by graphing it  a (x + b)² + c= 0 a- changes the stepness of the line. b- moves right ot left. Left= + Positive= - c- moves the vertex up or down. (Positive goes up. Negative goes down.) * If a is less than 0 if a is bigger than 0

Examples:  Y= -2 (x-4)²+5 Y= 2(x+3) 2-2

Y=4/2 (x-2) -6 Y=2/4(x+3) -3

How to solve quadratic equation using square roots  X²=k  If your equation has a # next to x: 1. You have to divide both sides by that # to isolate x². Then you simplify. Use the square root property to obtain to posible answers.

Examples: 1. K²=16 √ 16 k= 4, K²=21 √ 21 k= 7, n²= 20 4 N²=5,-5 4.7x² = x²= 3, -3

How to solve quadratic equation using factoring:  In order to factor a quadratic you must find common numbers that will multiply b and add up to c. Then put each set = 0. x 2 + 5x + 6 = (x + 2)(x + 3) (x + 2)(x + 3) = 0 x + 2 = 0 or x + 3 = 0 x = –2 or x = – 3 x = –3, –2 x 2 – 2x – 3 = 0 (x – 3)(x + 1) = 0 x – 3 = 0 or x + 1 = 0 x = 3 or x = –1 x = –1, 3

x 2 + 5x – 6 = 0 (x + 6)(x – 1) = 0 x + 6 = 0 or x – 1 = 0 x = –6 or x = 1 x = –6, 1 x 2 +5x+6=0. (x+2)(x+3)=0. x=-2 and x=-3..

Completing the square  To complete the square: 1. Get a=1 2. Find b, divide b/2, square it (b/2)² 3. Factor (x+b/2)² Ex. X² + 14x + 49 x²+26x+169

How to solve quadratic equations using completing the square:  STEPS: 1. get x²=1 2. get c by itself 3. comple the square 4. add b/2² to both sides 5. square root both sides

Examples: 1. A² + 2 a – 3= A²+ 2 a= 3 +1 √(a+1)² = √4 A + 1= ± 2 A= 3,1 2. A² - 2a – 8= 0 +8 A²- 2a= 8 +1 √(a-1)² = √9 A-1= ± 3 A= 4,2 3. X² + 16p – 22= X² + 16x= x+1= ± 4.8 X= 5.8, X² + 8k + 12 = X² + 8x = 13 X+1= ±3.6 X= 4.6, 2.6 √(x + 1)² =√23 √(x+1)² =√13

How to solve quadratic equations using quadratic formula:  X= -b ± √b²-4ac 2 a 1. Find a, b, c and fill them in. 3x² -4x -9= 0 A= 3 b= -4= c=9 4± √16+108= 4± √ ± √31 3 M²- 5m-14=0 A= 1 b=-5 c= ± √ = 5± √ ± √9 2

C²- 4c + 4= 0 4± √16-16 = 4± √ ± √0 -4 3± √9+40 = 3± √ ± √7 4