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 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 factorization method is a simple one.
I will state clearly in this article the easiest way to factorize 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 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 quadratic equation : When the coefficient of x2 is not 1
In this case the general formula becomes
ax2 + bx + c = 0
Follow the 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 step 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.
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