# How to Solve a Quadratic Equation

## Solving a Quadratic Equation

A quadratic equation is an equation in the form:

Solving a quadratic equation mean finding the value of **x** that makes this equation true (i.e. makes the left hand side equal to 0.)

The values of **x** that solve the equation are called the *roots* of the equation.

## Understanding Solving Quadratic Equations

It is easier to understand solving quadratic equations with an example. Let's look at a quadratic equation.

**x** is a variable. It can take different values.

Let's try **x = 1**, **x = 2** and **x = 3**.

### x = 1

Substitute **x = 1** into the left hand side of the quadratic equation:

x^{2} − 3x + 2 = (*1* )^{2} − 3(*1* ) + 2

x^{2} − 3x + 2 = *1* × *1* − 3 × *1* + 2

x^{2} − 3x + 2 = 1 − 3 + 2

x^{2} − 3x + 2 = 0

When **x = 1**, the left hand side of the equation equals **0**, which is equal to the right hand side of the equation.

**x = 1** solves the equation. It is a root of the equation.

### x = 2

Substitute **x = 2** into the left hand side of the quadratic equation:

x^{2} − 3x + 2 = (*2* )^{2} − 3(*2* ) + 2

x^{2} − 3x + 2 = *2* × *2* − 3 × *2* + 2

x^{2} − 3x + 2 = 4 − 6 + 2

x^{2} − 3x + 2 = 0

When **x = 2**, both sides of the equation are equal.

**x = 2** solves the equation. It is a root of the equation.

### x = 3

Substitute **x = 3** into the left hand side of the quadratic equation:

x^{2} − 3x + 2 = (*3* )^{2} − 3(*3* ) + 2

x^{2} − 3x + 2 = *3* × *3* − 3 × *3* + 2

x^{2} − 3x + 2 = 9 − 9 + 2

x^{2} − 3x + 2 = 2 ≠ 0

When **x = 3**, the left hand side of the equation equals **2**. This is *not* equal to the right hand side of the equation, **0**.

**x = 3** does not solve the equation.

**x = 1** and **x = 2** solve the quadratic equation **x ^{2} − 3x + 2 = 0**.

A quadratic equation will always have **2** values of **x** that solve the equation. There are always **2** roots.

## How to Solve Quadratic Equations

There are 3 ways to solve quadratic equations.

### Factoring

A quadratic equation can sometimes be written as the product of two brackets.

For example:

from this, we can read off the two roots of the quadratic equation:

**x = 1**, **x = 2**

### Quadratic Formula

A quadratic equation can be solved using the quadratic formula:

In this formula, **a**, **b** and **c** are the numbers in the quadratic equation in standard form, **ax ^{2} + bx + c**.

Read more about solving quadratic equations using the quadratic formula

### Graph

A quadratic equation can be solved by plotting it on a graph and finding where it crosses the x-axis:

In this graph above, the quadratic curve crosses the x-axis at **x = 1** and **x = 2**. These are the roots of the equation, that solve the equation.