Factorization of quadratic equations is the part of finding the roots of a quadratic equation. Factoring quadratic equations means converting the given quadratic expression into the product of two linear factors. Before understanding the factorization of quadratic equations, let’s recall what is a quadratic equation and its standard form. When a quadratic polynomial equates to 0, we get the quadratic equation. If ax2 + bx + c is the quadratic polynomial, ax2 + bx + c = 0 is the quadratic equation, where a, b, c are real numbers such that a ≠0. As the degree of quadratic equation 2, it contains two roots. In this article, you will learn the methods of solving quadratic equations by factoring, as well as examples with solutions.
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Factorization Method of Quadratic Equations
Factorization of quadratic equations can be done in different methods. They are:
- Splitting the middle term
- Using formula
- Using Quadratic formula
- Using algebraic identities
Let’s understand how to factor the given quadratic equation using all these methods.
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Factorization of Quadratic Equation by Splitting the Middle term
Step 1: Consider the quadratic equation ax2 + bx + c = 0
Step 2: Now, find two numbers such that their product is equal to ac and sum equals to b.
(number 1)(number 2) = ac
(number 1) + (number 2) = b
Step 3: Now, split the middle term using these two numbers,
ax2 + (number 1)x + (number 2)x + c = 0
Step 4: Take the common factors out and simplify.
Let’s have a look at the example problem given below:
| Question: Solve the quadratic equation x2 + 7x + 10 = 0 by splitting the middle term.
Solution: Given, x2 + 7x + 10 = 0 Here, a = 1, b = 7, c = 10 ac = (1)(10) = 10 Factors of 10: 1, 2, 5, 10 Let’s identify two factors such that their sum is 7 and the product is 10. Sum of two factors = 7 = 2 + 5 Product of these two factors = (2)(5) = 10 Now, split the middle term. x2 + 2x + 5x + 10 = 0 Take the common terms and simplify. x(x + 2) + 5(x + 2) = 0 (x + 5)(x + 2) = 0 Thus, (x + 2) and (x + 5) are the factors of the given quadratic equation. Solving these two linear factors, we get x = -2, -5 as roots. |
Also, check: Quadratic Equation Solver
Factoring Quadratic Equation using Formula
This method is almost similar to the method of splitting the middle term.
Step 1: Consider the quadratic equation ax2 + bx + c = 0
Step 2: Now, find two numbers such that their product is equal to ac and sum equals to b.
(number 1)(number 2) = ac
(number 1) + (number 2) = b
Step 3: Substitute these two numbers in the formula given below:
(1/a) [ax + (number 1)] [ax + (number 2)] = 0
Step 4: Finally simplify the above equation.
Go through the example given below to understand the above method in a better way.
| Question: Factorize 3x2 + 7x + 4 = 0
Solution: 3x2 + 7x + 4 = 0 Here, a = 3, b = 7, c = 4 ac = (3)(4) = 12 Let’s identify two numbers such that their sum is 7 and the product is 12. Factors of 12: 1, 2, 3, 4, 6, 12 Sum of two factors = 7 Product of those two factors = 12 Number 1 = 3 and number 2 = 4 Now, substitute these two numbers in the formula (1/a) [ax + (number 1)] [ax + (number 2)] = 0. (⅓) (3x + 3)(3x + 4) = 0 (3x + 3)(3x + 4) = 0 3(x + 1)(3x + 4) = 0 (x + 1)(3x + 4) = 0 Thus, (x + 1) and (3x + 4) are the factors of the given quadratic equation. By solving these, we get x = -1, -4/3 as roots. |
Factoring Quadratic Equation using Quadratic Formula
Suppose p and q are the factors of a quadratic equation in x, then (x – p) and (x – q) will be the factors of it. Keeping this in mind, we can factorize the quadratic equation using the quadratic formula. Using the quadratic formula, we can get the roots of a quadratic equation, and using these roots we can write the factors.
Quadratic formula to get the roots of a quadratic equation ax2 + bx + c = 0 is given by:
x = [-b ± √(b2 – 4ac)]/ 2a
Substituting the values of a, b, c and simplifying the expression, we get the values of roots.
This can be better understood with the help of an example given below:
| Question: Factorize x2 + 4x – 21 = 0 using quadratic formula.
Solution: Given, x2 + 4x – 21 = 0 Here, a = 1, b = 4, c = -21 b2 – 4ac = (4)2 – 4(1)(-21) = 16 + 84 = 100 Substituting these values in the quadratic formula, we get; x = [-4 ± √100]/ 2(1) = (-4 ± 10)/2 x = (-4 + 10)/2, x = (-4 – 10)/2 x = 6/2, x = -14/2 x = 3, x = -7 Therefore, the factors of the given quadratic equation are (x – 3) and (x + 7). |
Factoring Quadratic Equations using Algebraic Identities
Two algebraic identities can be applied to factor the given quadratic equation. This method can be applied only when the LHS of the given quadratic equation is in the form a2 + 2ab + b2 or a2 – 2ab + b2.
We know that:
- a2 + 2ab + b2 = (a + b)2 = (a + b)(a + b)
- a2 – 2ab + b2 = (a – b)2 = (a – b)(a – b)
- a2 – b2 = (a + b)(a – b)
Below are the solved examples of factoring the quadratic equation using identities.
| Factorize: x2 + 6x + 9 = 0 | Factorize: x2 – 10x + 25 = 0 | Factorize: 9x2 – 16 = 0 |
| x2 + 6x + 9 = 0
This can be written as: (1x)2 + 2(1)(3)x + (3)2 = 0 LHS is of the form a2 + 2ab + b2, ⇒ (x + 3)2 = 0 Or (x + 3)(x + 3) = 0 Factors are (x + 3) and (x + 3). |
x2 – 10x + 25 = 0
This can be written as: (1x)2 – 2(1)(5)x + (5)2 = 0 LHS is of the form a2 – 2ab + b2, ⇒ (x – 5)2 = 0 Or (x – 5)(x – 5) = 0 Factors are (x – 5) and (x – 5). |
9x2 – 16 = 0
This can be written as: (3x)2 – (4)2 = 0 LHS is of the form a2 – b2, ⇒ (3x + 4)(3x – 4) = 0 Factors are (3x + 4) and (3x – 4). |
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