Factorization of Quadratic Trinomials is the process of finding factors of given Quadratic Trinomials. If ax^2 + bx + c is an expression where a, b, c are constants, then the expression is called a quadratic trinomial in x. The expression ax^2 + bx + c has an x^2 term, x term, and an independent term. Find Factoring Quadratics Problems with Solutions in this article.

## Factorization of Quadratic Trinomials Forms

The Factorization of Quadratic Trinomials is in two forms.

(i) First form: x^2 + px + q

(ii) Second form: ax^2 + bx + c

### How to find Factorization of Trinomial of the Form x^2 + px + q?

If x^2 + px + q is an Quadratic Trinomial, then x^{2} + (m + n) × + mn = (x + m)(x + n) is the identity.

**Solved Examples on Factorization of Quadratic Trinomial of the Form x^2 + px + q **

1. Factorize the algebraic expression of the form x^2 + px + q

(i) a^{2} – 7a + 12

Solution:

The Given expression is a^{2} – 7a + 12.

By comparing the given expression a^{2} – 7a + 12 with the basic expression x^2 + px + q.

Here, a = 1, b = – 7, and c = 12.

The sum of two numbers is m + n = b = – 7 = – 4 – 3.

The product of two number is m * n = a * c = -4 * (- 3) = 12

From the above two instructions, we can write the values of two numbers m and n as – 4 and -3.

Then, a^{2} – 7a + 12 = a^{2} – 4a -3a + 12.

= a (a – 4) – 3(a – 4).

Factor out the common terms.

Then, a^{2} – 7a + 12 = (a – 4) (a – 3).

(ii) a^{2} + 2a – 15

Solution:

The Given expression is a^{2} + 2a – 15.

By comparing the given expression a^{2} + 2a – 15 with the basic expression x^2 + px + q.

Here, a = 1, b = 2, and c = -15.

The sum of two numbers is m + n = b = 2 = 5 – 3.

The product of two number is m * n = a * c = 5 * (- 3) = -15

From the above two instructions, we can write the values of two numbers m and n as 5 and -3.

Then, a^{2} + 2a – 15 = a^{2} + 5a – 3a – 15.

= a (a + 5) – 3(a + 5).

Factor out the common terms.

Then, a^{2} + 2a – 15 = (a + 5) (a – 3).

### How to find Factorization of trinomial of the form ax^2 + bx + c?

To factorize the expression ax^2 + bx + c we have to find the two numbers p and q, such that p + q = b and p × q = ac

**Solved Examples on Factorization of trinomial of the form ax^2 + bx + c **

2. Factorize the algebraic expression of the form ax2 + bx + c

(i) 15b^{2} – 26b + 8

Solution:

The Given expression is 15b^{2} – 26b + 8.

By comparing the given expression 15b^{2} – 26b + 8 with the basic expression ax^{2} + bx + c.

Here, a = 15, b = -26, and c = 8.

The sum of two numbers is p + q = b = -26 = 5 – 3.

The product of two number is p * q = a * c = 15 * (8) = 120

From the above two instructions, we can write the values of two numbers p and q as -20 and -6.

Then, 15b^{2} – 26b + 8 = 15b^{2} – 20 – 6b + 8.

= 5b (3b – 4) – 2(3b – 4).

Factor out the common terms.

Then, 15b^{2} – 26b + 8 = (3b – 4) (5b – 2).

(ii) 3a^{2} – a – 4

Solution:

The Given expression is 3a^{2} – a – 4.

By comparing the given expression 3a^{2} – a – 4 with the basic expression ax^{2} + bx + c.

Here, a = 3, b = -1, and c = -4.

The sum of two numbers is p + q = b = -1 = 5 – 3.

The product of two number is p * q = a * c = 3 * (-4) = -12

From the above two instructions, we can write the values of two numbers p and q as -4 and 3.

Then, 3a^{2} – a – 4 = 3a^{2} – 4a -3a – 4.

= a (3a – 4) – 1(3a – 4).

Factor out the common terms.

Then, 3a^{2} – a – 4 = (3a – 4) (a – 1).