# Factoring Trinomials Formula

Before going to learn the factoring trinomials formulas, let us recall what is a trinomial. A trinomial is a polynomial with three terms. Factoring means writing an expression as the product of two or more expressions. A trinomial can be a perfect square or a non-perfect square. Let us learn the factoring trinomials formulas along with a few solved examples.

## What Are Factoring Trinomials Formulas?

A trinomial can be a perfect square or a non-perfect square. We have two formulas to factorize a perfect square trinomial. But for factorizing a non-perfect square trinomial, we do not have any specific formula, instead, we have a process.

- The factoring trinomials formulas of perfect square trinomials are:
a

^{2}+ 2ab + b^{2}= (a + b)^{2}a

^{2}- 2ab + b^{2}= (a - b)^{2}For applying either of these formulas, the trinomial should be one of the forms a

^{2}+ 2ab + b^{2}(or) a^{2}- 2ab + b^{2}. - The process of factoring a non-perfect trinomial ax
^{2}+ bx + c is:Step 1: Find ac and identify b.

Step 2: Find two numbers whose product is ac and whose sum is b.

Step 3: Split the middle term as the sum of two terms using the numbers from step - 2.

Step 4: Factor by grouping.

Let us see the applications of factoring trinomials formulas in the following section.

## Examples Using Factoring Trinomials Formulas

**Example 1:** Factor x^{2} - 16x + 64 using the factoring trinomials formulas.

**Solution:**

We can write the given trinomial as,

x^{2} - 16x + 64 = x^{2} - 2(x)(8) + 8^{2}

The right side trinomial is of the form a^{2} - 2ab + b^{2} and hence we can apply the formula,

a^{2} - 2ab + b^{2} = (a - b)^{2}

Thus, x^{2} - 2(x)(8) + 8^{2} = (x - 8)^{2}

**Answer: **x^{2} - 16x + 64 = (x - 8)^{2}.

**Example 2:** Factor the trinomial 2x^{2} - x - 3.

**Solution:**

We use the factoring trinomial formula of non-perfect trinomials to factor the given trinomial.

Comparing 2x^{2} - x - 3 with ax^{2} + bx + c, we get a = 2, b = -1, and c = -3.

Here ac = 2(-3) = -6 and b = -1.

Two numbers whose product is -6 and whose sum is -1 are -3 and 2.

We will split the middle term -x as -3x + 2x and then we factor by grouping the terms.

2x^{2} - x - 3

= 2x^{2} - 3x + 2x - 3

= x (2x - 3) + 1 (2x - 3)

= (2x - 3)(x + 1)

**Answer:** 2x^{2} - x - 3 = (2x - 3)(x + 1).

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