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
Solving quadratic equations by factoring

Relevant for

Learning to solve quadratic equations by factoring.

See examples

ALGEBRA
Solving quadratic equations by factoring

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.

If you need to learn or reinforce the techniques for factoring quadratic equations, you can visit our article: Factoring Quadratic Equations.


Solve quadratic equations by factoring – Examples with answers

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.

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

Choose an answer






What are the solutions of the equation $latex x^2-3x-4=0$?

Choose an answer






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

Choose an answer






What are the solutions of the equation $latex 3x^2-7x+2=0$?

Choose an answer






Find the solutions of the equation $latex 2x^2-3x-5=0$.

Choose an answer







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