Start with the case where a=1, and try to use it if possible.
See solution.
Practice makes perfect
Let's review how to factor a expression of the form x^2+bx+c first, then we will move to one of the form ax^2+bx+c when a > 1.
Factoring Expressions of the Form x^2+bx+c
Let's factor the expression x^2+2x-24.
x^2+bx+c
x^2+ 2x- 24
To factor this expression we need to find the factors of c such that their sum is b. For this case, we need numbers whose product is 24 and sum is 2. Since -24<0, the factors should have different signs. And, since 2>0, the greatest factor should be positive. Let's check these requirements for different factors of -24.
Factors of - 24
Sum
- 1,24
-1 +24 = 23 *
-2,12
-2 + 12 = 10 *
- 3,8
- 3 + 8 = 5 *
-4,6
-4 + 6= 2 ✓
With this in mind, we can factor our original expression.
x^2+2x-24 =3 (x- 4)(x+ 6)
Factoring Expressions of the Form ax^2+bx+c When a > 1
For expressions of this form it is recommended to start by looking for the greatest common factor (GCF). This is the common factor with the greatest coefficient and the greatest exponent. Let's consider the case 3x^2 + 6x-72, for instance. In this case the GCF is 3.
3x^2+6x-72 & = 3(x^2) + 3(2x)-3(24)
3x^2+6x-72 & =3 (x^2+ 2x- 24)
We can see that the expression inside the parentheses is the same from the previous case. However, to reduce a expression of the more general form ax^2+bx+c to one of the form x^2 +bx+c is not always possible. If we consider 2x^2+11x+12, for instance, the GCF is 1. In that case we have to proceed differently.
ax^2+bx+c
2 x^2+11x+12
We need to look for factors of ac such that their sum is b. In this case, ac=(2)(12) = 24. Since 24>0 we will need the factors to have the same sign. And, since 11>0, we know both should be positive. Let's check for these requirements for different factors of 24.
Factors of 24
Sum
1,24
1 + 12 = 11 *
2,12
2 + 6 = 4 *
3,8
3 + 8 = 11 ✓
4,6
3 + 4 = *
Now we rewrite our original expression using the factors that met the requirements mentioned before.
2x^2+11x +12 = 2x^2+3x+8x +12
We continue by looking for the GCF of the first two terms and the two last terms.
2x^2+3x+8x +12 = x(2x+3)+4(2x+3)
Finally, notice that both expressions have as a common factor the binomial (2x+3). We can factor the expression once more.
x(2x+3)+4(2x+3) = (2x+3)(x+4)
We found that the expression 2x^2+11x+12 can be factored as (2x+3)(x+4).
