Download 5.3 Factoring Quadratic Function

5.4
Factoring ax2 + bx +c
12/10/2012
In the previous section we learned to factor x2 + bx + c
where a = 1.
In this section, we’re going to factor ax2 + bx + c where
a ≠ 1.
Ex: Factor 3x2 + 7x +2
We’re still going to use the “Big X” Method.
The Big “X” method
Simplify
like a
fraction
if needed
a
multiply
a
a•c
#1
#2
Factor: ax2 + bx + c
Think of 2 numbers
that Multiply to a•c
and Add to b
Simplify like a
fraction if
needed
b
add
Answer: Write the simplified answers in
the 2 ( ). Top # is coefficient of x and
bottom # is the 2nd term
Factor: 3x2 + 7x + 2
1 3
Simplify
like a
fraction .
÷ by 3
3•2 =
6
6
2
3
1
Think of 2 numbers that
Multiply to 6 and Add to 7
6x1=6
6+1=7
7
multiply
a
a•c
a
#1
Answer: (1x + 2) (3x + 1)
(x + 2) (3x + 1)
or
#2
b
add
Checkpoint
Factor ax 2 + bx + c when c is Positive
Factor the expression.
1. 2x 2 + 11x + 5
ANSWER
( 2x + 1 ) ( x + 5 )
2. 2y 2 + 9y + 7
ANSWER
( 2y + 7 ) ( y + 1 )
3. 3r 2 + 8r + 5
ANSWER
( 3r + 5 ) ( r + 1 )
Factor: 4x2 - 16x - 9
2
Simplify
like a
fraction .
÷ by 2
4
4(-9) =
42
-36
-18
-9
2
1
Think of 2 numbers that
Multiply to -36 and
Add to -16
-18 x 2 = -36
-18 + 2 = -16
Simplify
like a
fraction .
÷ by 2
-16
multiply
a
a•c
a
#1
Answer: (2x - 9) (2x + 1)
#2
b
add
Factor: 6x2 + 27x - 15
Do 6, 27 and -15 have any factors in common?
Yes, 3. Factor 3 out.
3(2x2 + 9x – 5). Then Factor what’s in the ( ). Think of 2 numbers
that Multiply to -10
and Add to 9
-1 x 10 = -10
2(-5) =
2
2
1 Simplify
-1 + 10 = 9
-10
-1
10
5
like a
fraction .
÷ by 2
9
a
multiply
a
a•c
#2
#1
b
add
Answer: 3(2x - 1) (x + 5)
(Don’t forget the 3!!!)
Checkpoint
Factor ax 2 + bx + c
Factor the expression.
4. 6z 2 + z – 12
ANSWER
( 3z – 4) ( 2z + 3)
5. 11x 2 + 17x + 6
ANSWER
( 11x + 6) ( x + 1)
6. 4w 2 – 6w + 2
ANSWER
2 ( 2w – 1) ( w – 1)
Finding the Zeros of the Function
Is the same as solving
ax2+bx+c = 0
Graphically, finding the zeros of the quadratic
function means finding the x-intercepts of the
parabola.
Example 4
Find the Zeros of a Quadratic Function
Find the zeros of y = 3x 2 – x – 4.
SOLUTION
To find the zeros of the function, let
y = 0. Then solve for x.
y = 3x 2 – x – 4
Write original function.
0 = 3x 2 – x – 4
Let y = 0.
0 = ( 3x – 4 ) ( x + 1 )
Factor the right side.
3x – 4 = 0
x =
4
3
or
x +1 = 0
x = –1
Use the zero product property.
Solve for x.
Example 4
Find the Zeros of a Quadratic Function
ANSWER
The zeros of the function are
4
and – 1.
3
CHECK The zeros of a function are
also the x-intercepts of the
graph of the function. So,
the answer can be checked
by graphing y = 3x 2 – x – 4.
The x-intercepts of the
graph are 4 and – 1, so the
3
answer is correct.
Checkpoint
Find the Zeros of a Quadratic Function
Find the zeros of the function.
7. y =
3x 2
8. y =
2x 2
9. y =
4x 2
– 2x – 1
– 7x + 3
– 18x + 8
ANSWER
1
– ,1
3
ANSWER
1
,3
2
ANSWER
1
,4
2
Homework
5.4 p.244 #18-25, 46-48, 57-59