Upload
tuan
View
37
Download
0
Tags:
Embed Size (px)
DESCRIPTION
QUADRATICS JOURNAL. Amani Mubarak 9-5. How to factor polynomials. 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. - PowerPoint PPT Presentation
Citation preview
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