Step by step explanation:
Example:
Multiplying (x+4) and (x−1) together (called Expanding) gets x2 + 3x − 4 :
expand vs factor quadratic
So (x+4) and (x−1) are factors of x2 + 3x − 4
Just to be sure, let us check:
(x+4)(x−1) = x(x−1) + 4(x−1)
= x2 − x + 4x − 4
= x2 + 3x − 4 yes
Yes, (x+4) and (x−1) are definitely factors of x2 + 3x − 4
Did you see that Expanding and Factoring are opposites?
expand vs factor quadratic
Common Factor
First check if there any common factors.
Example: what are the factors of 6x2 − 2x = 0 ?
6 and 2 have a common factor of 2:
2(3x2 − x) = 0
And x2 and x have a common factor of x:
2x(3x − 1) = 0
And we have done it! The factors are 2x and 3x − 1,
We can now also find the roots (where it equals zero):
2x is 0 when x = 0
3x − 1 is zero when x = 13
And this is the graph (see how it is zero at x=0 and x= 13 ):
graph of 6x^2 - 2x
Example: what are the factors of 2x2 + 7x + 3 ?
No common factors.
We could guess (2x+3)(x+1):
(2x+3)(x+1) = 2x2 + 2x + 3x + 3
= 2x2 + 5x + 3 (WRONG)
How about (2x+7)(x−1):
(2x+7)(x−1) = 2x2 − 2x + 7x − 7
= 2x2 + 5x − 7 (WRONG AGAIN)
how about (2x+9)(x−1):
(2x+9)(x−1) = 2x2 − 2x + 9x − 9
= 2x2 + 7x − 9 (WRONG AGAIN)
A Method For Simple Cases
With the quadratic equation in this form:
Quadratic Equation
Step 1: Find two numbers that multiply to give ac (in other words a times c), and add to give b.
Example: 6x2 + 5x − 6
Step 1: ac is 6×(−6) = −36, and b is 5
List the positive factors of ac = −36: 1, 2, 3, 4, 6, 9, 12, 18, 36
One of the numbers has to be negative to make −36, so by playing with a few different numbers I find that −4 and 9 work nicely:
−4×9 = −36 and −4+9 = 5
Step 2: Rewrite 5x with −4x and 9x:
6x2 − 4x + 9x − 6
Step 3: Factor first two and last two:
2x(3x − 2) + 3(3x − 2)
Step 4: Common Factor is (3x − 2):
(2x+3)(3x − 2)
