# Factoring Trinomials Using Quadratic Equation Roots

For factorization of trinomials, there are several different ways including:

Factoring by grouping *(will be discussed later)*;

Factoring by completing the square *(will be discussed later)*;

Factoring a simple trinomial;

Factoring with the leading coefficient;

But in this lesson, we will show another method, which may be the best in the event that it is not possible to use the other methods, and it may also be the easiest.

This method depends on transforming the trinomial into a quadratic equation, then we solve this equation as explained in the lesson "Solving of a Quadratic Equations". When we find the solutions to the equation, we can then infer factoring the trinomial.

The full method will be explained, just follow the article.

To factor a trinomial, first we must transform it to a standard form of a quadratic equation.

### Note:

Given a trinomial: with .

The equivalent quadratic equation is:

In the standard form, there is always 0 on the right side, and the leading coefficient must be different from 0.

To solve this equation, we must calculate the discriminant of the equation.

The quantity can be positive, negative, or equal to 0.

If , so, there are two real roots;

If , so, there is one real root;

If , so, there are no real roots;

The note bellow show you how to factor the trinomial according to the roots.

### Note:

Let be a quadratic equation.

- If . The equation has two different real roots:

Rewriting the trinomial as a factoring form:

- If . The equation has one real root:

Rewriting the trinomial as a factoring form:

- If . The equation has one real root. Then the trinomial cannot be factored.

Note that is the leading coefficient of the trinomial.

In these examples we will try to take some examples from the three cases to factor a trinomials using quadratic formula.

**Example1:**

Factor the trinomial:

**Solution:**

- The equivalent quadratic equation is:

- Calculate the discriminant:

- The discriminant is positive, so, there are two real roots:

- The trinomial can be factored as follow:

**Example2:**

Factor the trinomial:

**Solution:**

- The equivalent quadratic equation is:

- Calculate the discriminant:

- The discriminant is positive, so, there are two real roots:

- The trinomial can be factored as follow:

**Example3:**

Factor the trinomial:

**Solution:**

- The equivalent quadratic equation is:

- Calculate the discriminant:

- The discriminant is equal to zero, so, there is one real root:

- The trinomial can be factored as follow:

**Example4:**

Factor the trinomial:

**Solution:**

- The equivalent quadratic equation is:

- Calculate the discriminant:

- The discriminant is equal to zero, so, there is one real root:

- The trinomial can be factored as follow:

**Example5:**

Factor the trinomial:

**Solution:**

- The equivalent quadratic equation is:

- Calculate the discriminant:

- The discriminant is negative, so, there are no real roots, and the trinomial cannot be factored.