How to Solve a Quadratic Equation

Solving a Quadratic Equation

A quadratic equation is an equation in the form:

y equals a x squared plus b x plus c

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.

2 x squared plus 3 x plus 4 equals 0

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:

x2 − 3x + 2 = (1 )2 − 3(1 ) + 2

x2 − 3x + 2 = 1 × 1 − 3 × 1 + 2

x2 − 3x + 2 = 1 − 3 + 2

x2 − 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:

x2 − 3x + 2 = (2 )2 − 3(2 ) + 2

x2 − 3x + 2 = 2 × 2 − 3 × 2 + 2

x2 − 3x + 2 = 4 − 6 + 2

x2 − 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:

x2 − 3x + 2 = (3 )2 − 3(3 ) + 2

x2 − 3x + 2 = 3 × 3 − 3 × 3 + 2

x2 − 3x + 2 = 9 − 9 + 2

x2 − 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 x2 − 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:

x squared minus 3 x plus 2 equals (x minus 1) (x minus 2)

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

x = 1, x = 2

Read more about solving quadratic equations using factoring

Quadratic Formula

A quadratic equation can be solved using the quadratic formula:

x equals minus b plus or minus the square root of b squared minus 4 a c all divided by 2 a

In this formula, a, b and c are the numbers in the quadratic equation in standard form, ax2 + 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:

the quadratic curve crosses the x-axis at x = 1 and x = 2

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.

Read more about solving quadratic equations using a graph

See Also

What is a quadratic equation? What is an equation? What is the x-axis? Graph a quadratic equation ( interactive widget)