# Solving Quadratic Equations by Factoring

A quadratic equation of the form ax²+bx+c=0 can be solved using the factorization method. For this, we have to factor the equation using whatever method is applicable to write it in the form (x+p)(x+q)=0. By forming an equation with each factor, we will find that the roots of the quadratic equation are x=-p and x=-q.

In this article, we will learn how to solve quadratic equations using the factoring method. In addition, we will use this method to solve some practice problems.

##### ALGEBRA

Relevant for

Learning to solve quadratic equations by factoring.

See examples

##### ALGEBRA

Relevant for

Learning to solve quadratic equations by factoring.

See examples

## How to solve quadratic equations by factoring?

To solve a quadratic equation by the factorization method, we have to follow the following steps:

Step 1: Simplify and write the equation in the form $latex ax^2+bx+c=0$.

Step 2: Factor the quadratic equation using any method so that we can write it in the form $latex (x+p)(x+q)=0$.

Step 3: Form an equation with each factor by setting it equal to zero. For example, $latex x+p=0$.

Step 4: Solve the equation for each factor.

Recall that factoring a quadratic equation consists of writing an equation from the form $latex x^2+bx+c=0$ to the form $latex (x+p)(x+q)=0$. To achieve this, we have to find two factors, which when multiplied, result in the original quadratic equation.

For example, the equation $latex x^2+2x-3=0$ can be factored in the form $latex (x+3)(x-2)=0$, since multiplying the factors gives us the original equation.

The following examples are solved by applying the factorization method. Try to solve the problems yourself before looking at the solution.

### EXAMPLE 1

Solve the equation $latex x^2+5x+6=0$.

Factoring the left-hand side of the equation, we have:

$latex x^2+5x+6=0$

$latex (x+2)(x+3)=0$

$latex x+2=0~~$ or $latex ~~x+3=0$

$latex x=-2~~$ or $latex ~~x=-3$

The solutions of the equation are $latex x=-2$ and $latex x=-3$.

### EXAMPLE 2

Find the solutions of the equation $latex x^2+2x-8=0$.

We are going to factor the left-hand side of the equation and then form equations with the factors to find the solutions:

$latex x^2+2x-8=0$

$latex (x+4)(x-2)=0$

$latex x+4=0~~$ or $latex ~~x-2=0$

$latex x=-4~~$ or $latex ~~x=2$

The solutions of the equation are $latex x=-4$ and $latex x=2$.

### EXAMPLE 3

Solve the equation $latex 2x^2-13x-24=0$ using the factoring method.

Factoring the left-hand side of the equation, we have:

$latex 2x^2-13x-24=0$

$latex (2x+3)(x-8)=0$

$latex 2x+3=0~~$ or $latex ~~x-8=0$

$latex x=-\frac{3}{2}~~$ or $latex ~~x=8$

The solutions of the equation are $latex x=-\frac{3}{2}$ and $latex x=8$.

### EXAMPLE 4

Solve the equation $latex x^2-x-10=x+5$ using the factoring method.

First, we need to simplify and write the equation in the form $latex ax^2+bx+c=0$. Then, we factor it and find its roots:

$latex x^2-x-10=x+5$

$latex x^2-2x-15=0$

$latex (x+3)(x-5)=0$

$latex x+3=0~~$ or $latex ~~x-5=0$

$latex x=-3~~$ or $latex ~~x=5$

The solutions are $latex x=-3$ and $latex x=-5$.

### EXAMPLE 5

Use the factoring method to solve the equation $latex 3x^2-10x+3=0$.

We factor the left-hand side of the equation, form an equation with each factor, and solve:

$latex 3x^2-10x+3=0$

$latex (3x-1)(x-3)=0$

$latex 3x-1=0~~$ or $latex ~~x-3=0$

$latex x=\frac{1}{3}~~$ or $latex ~~x=3$

The roots of the equation are $latex x=\frac{1}{3}$ and $latex x=3$.

### EXAMPLE 6

Find the solutions of the equation $latex 5x^2 -5x-10=0$.

We can start by dividing both sides of the equation by 5 to simplify it. Then, we factor the left-hand side and solve for the factors:

$latex 5x^2-5x-10=0$

$latex x^2-x-2=0$

$latex (x+1)(x-2)=0$

$latex x+1=0~~$ or $latex ~~x-2=0$

$latex x=-1~~$ or $latex ~~x=2$

The solutions of the equation are $latex x=-1$ and $latex x=2$.

### EXAMPLE 7

Solve the equation $latex 3x^2+14x-12=2x^2+15x$ using the factoring method.

To factor this equation, we need to start by simplifying it and write it in the form $latex ax^2+bx+c=0$. Then, we factor it and solve for the factors:

$latex 3x^2+14x-12=2x^2+15x$

$latex x^2-x-12=0$

$latex (x+3)(x-4)=0$

$latex x+3=0~~$ or $latex ~~x-4=0$

$latex x=-3~~$ or $latex ~~x=4$

The solutions of the equation are $latex x=-3$ and $latex x=4$.

## Solve quadratic equations by factoring – Practice problems

Use the factoring method to find the solutions of the following quadratic equations.