QUADRATICS JOURNAL

Amani Mubarak 9-5

How to factor polynomials

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 -62 2

-2 + -6

Ex.1 12 8

X= -1, -3

3x²-2x-16= 0 -48 -2

-8X6 -

8+6

3 3

X= 8/3, -2

Ex.4

Ex.3

Ex.2

x² +7x +15= 5 10 7 -5 2x5 2+5

x² + 7x + 15= 0 x x= 2,5

x² + 8x = -15

+15 +15

X²+8x+15= 0

15 8

3X5 3+5

x

x= 3,5

QUADRATIC FUNCTION A Quadratic Function is written in the

form: f(x) = ax2 + 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.

Ex.1

y= 1/2x + 2

Ex.2

y=3x² + 6x + 1

Ex. 3 Ex. 4

y= -x + 5

y= 1/2x² +0 + 0

HOW TO GRAPH A QUADRATIC FUNCTION f (x) = ax2 + bx + c

Y intercept of the graph is found by f(0)=c X intercept of the graph is found by solving the

equation: ax2 + bx + c = 0 ax2 + 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 + 0 + 0

-b/2ª= 0

x y

0 0

1 -1

2 -4

3 -9

X=0,0

2. y= 1/2x² +0 + 0

-b/2(a)=0

x y

0 0

1 0.5

2 2

3 4.5

X=0,0

3. y=3x² + 6x + 1

-b/2(a)= -6/2(3)

-6/6= -1

X Y

-1 -2

0 1

1 10

2 19

X= -1,-2

4. y= x² + 2x + 5

-b/2(a)= -2/2(0)= -2/2

x y

-2 -3

-1 -2

0 -1

1 0

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, -4

2. K²=21

√ 21

k= 7,-7

3. 4n²= 20

4

N²=5,-5

4.7x² = -21 7x²= 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.

Ex.1 x2 + 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

Ex.2x2 – 2x – 3 = 0

(x – 3)(x + 1) = 0

x – 3 = 0  or  x + 1 = 0 x = 3  or  x = –1 x = –1, 3

Ex.3

Ex.4

x2 + 5x – 6 = 0

(x + 6)(x – 1) = 0

x + 6 = 0  or   x – 1 = 0 x = –6  or   x = 1 x = –6, 1

x2+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= 0

+3 +3

A²+ 2 a= 3 +1

√(a+1)² = √4

A + 1= ± 2

A= 3,1

2. A² - 2a – 8= 0

+8 +8

A²- 2a= 8 +1

√(a-1)² = √9

A-1= ± 3

A= 4,2

3. X² + 16p – 22= 0

+22 +22

X² + 16x= 22 +1

x+1= ± 4.8

X= 5.8, 3.8

4. X² + 8k + 12 = 0

-12 -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.

Ex.1 3x² -4x -9= 0

A= 3 b= -4= c=9

4± √16+108= 4± √124 6 6 2± √31 3

Ex.2M²- 5m-14=0

A= 1 b=-5 c= -14

5 ± √ 25 + 56 = 5± √81 2 25± √9 2

Ex.3

Ex.4

C²- 4c + 4= 0

4± √16-16 = 4± √0 -8 -8

2± √0 -4 3± √9+40 = 3± √49

4 4

3± √7 4