A quadratic equation is an expression written in the form ax2 + bx + c = 0 in which a, b, c are numerals and the power of x is 2. This expression can be factorized by using the factorization of quadratic equation rules.
Note: The power of x can neither be fraction nor negative.
Factoring Quadratic Equation
The process of solving quadratic equations using the factorization method is a simple one.
I will state clearly in this article the easiest way to factorize the quadratic equation. Happy learning.
How Do I Know if a quadratic equation can be Factorized?
To check if a quadratic equation can be factored, use the discriminant b2 – 4ac. If it gives a perfect square, it can be factorized but if it does not, it cannot be factorized.
We shall consider two cases in order to give you a clear picture of how simple it is to factorize.
Case 1 for factorization of a quadratic equation: When the coefficient of x2 is 1 ( that is to say that a = 1)
The general quadratic equation now becomes
x2 + bx + c = 0
Steps to Factorize Quadratic Equation
- Check if the expression can be factored
- Check the sign of b and c
- If c is positive, get two factors of c whose product gives c and sum gives b. Both factors will have the sign of b.
- If c is negative, look for two factors of c whose product gives c and difference gives b. The numerically larger among the two factors will carry the sign of b and the smaller factor will carry the opposite sign.
- Replace the middle term with the factors
- Factorize by grouping the factors.
x2 + 5x + 6 = 0
a = 1, b = 5, c = 6
Checking to know if the expression can be factored
b2 – 4ac = 5×5 – 4x1x6
= 25 – 24 = 1
SQRT (1) = 1. It gives a perfect square, so it can be factored.
Let’s move to the next step
You can see here that the coefficient of x2 is 1
The signs of c and b are positive
Possible pairs of factors of 6 are (1, 6) and (2, 3)
since c is positive, both factors will have the same sign as b and here b is positive. The factors whose product is 6 and sum yields b are (2, 3)
Replacing the middle term 5x with the factors
x2 + 2x + 3x + 6 = 0
(x2 + 2x) (3x + 6)
x(x + 2) + 3(x+2)
As simple as that.
Case 2 for factorization of a quadratic equation: When the coefficient of x2 is not 1
In this case the general formula becomes
ax2 + bx + c = 0
Follow step 1 above. After step 1, get the product of a and c = ac (the product must be taken as positive regardless of the sign).
write down all the possible pairs of factors of ac. If c is positive, get two factors of ac whose product gives ac and sum gives b. Both factors should have the sign of b.
If c is negative, look for two factors of ac whose product gives ac and difference gives b. The numerically larger among the two factors will carry the sign of b and the smaller factor will carry the opposite sign.
Then, follow steps 5 to 6.
Factorize 6x2 + 11x + 3
Let’s get working.
a = 6, b = 11, c = 3
ac = 6×3 = 18
Possible pairs of factors of 18 are (1,18), (2, 9) and (3, 6)
c is positive.
Therefore, the two factors will carry the sign of b and their sum should give b
2+9 = 11.
This implies that 2 and 9 are the pairs of factors that satisfied the condition.
Let’s replace the middle term ’11x’ with the two factors
6x2 + 2x + 9x + 3 = 0
factoring by grouping
( 6x2 + 2x )( 9x + 3 )
Taking out the common factors
2x(3x + 1) + 3(3x+1)
(2x + 3)(3x+1)
That is it.
I hope you enjoyed it. Please use the comment section if you have any questions and remember to subscribe to more lessons.
You can share it with your friends using the social media icons in this post. Thanks for your time and do come back for more solutions.