Quadratic Equations

What Is a Quadratic Equation?

A quadratic equation is a type of equation. A quadratic equation is in the form:

a, b and c are constants that stand in for particular numbers. x is a variable and can take any number. The highest power of x in a quadratic equation is 2. It has an x2 term. Here is an example of a quadratic equation:

A quadratic equation shows a curve when plotted on a graph. Here is the quadratic equation above plotted on a graph:

Dictionary Definition

The Oxford English Dictionary defines a quadratic equation as "an equation of the second degree; specifically an equation of the form ax2 + bx + c = 0, where a, b, and c are constants and x is unknown."

Where Does the Word Quadratic Come From?

The word "quadratic" comes from the word "quad", meaning "square" - because the x is squared.

The Curriculum

The lessons are grouped into mini-curriculum to help you organise your learning. A brief description is given for each mini-curriculum. Click the MORE button to learn more.

Quadratic Equations

Quadratic equations are equations where x2 is the highest power of x. Quadratic equations show curves when plotted. In this mini-curriculum, you will learn about quadratic equations.

Quadratic Equation
A quadratic equation is an equation in the form:

Quadratic equations show a curve when they are plotted on a graph:


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Read more about quadratic equations

Solving Quadratic Equations

Solving a quadratic equation means finding the value of x that make the equation equal to 0. There are different ways to solve quadratic equations. In this mini-curriculum, you will learn how to solve quadratic equations.

Solve a Quadratic Equation
To solve a quadratic equation, find the value of x that makes the equation equal to 0:

Solve a Quadratic Equation Using Factoring
To solve a quadratic equation using factoring:

(1) Factor the Equation
Write the equation as two brackets multiplying each other.
Here is an example of factoring a quadratic equation into two brackets multiplying each other:

x2 + 5x + 4 = 0
⇒ (x + 1)(x + 4) = 0

(2) Solve the Equation
Equate each bracket to 0 to find the two values of x that solve the equation.
Here is an example of equating each bracket to solve the equation:

(x + 1)(x + 4) = 0
x + 1 = 0 ⇒ x = −1
x + 4 = 0 ⇒ x = −4

Solve a Quadratic Equation Using Factoring Where Leading Coefficient Is Not 1
A quadratic equation where there is a number before the x2 can be solved using factoring.
Here is an example of factoring and solving a quadratic equation where the leading coefficient is not 1:

(1) Factor the Equation
2x2 + 7x + 6 = 0
⇒ (x + 2)(2x + 3) = 0

(2) Solve the Equation

x + 2 = 0 ⇒ x = −2
2x + 3 = 0 ⇒ x = −32

Solve a Quadratic Equation Using the Quadratic Formula
To solve a quadratic equation using the quadratic formula, compare the quadratic equation with ax2 + bx + c = 0 to find the values of a, b and c. Substitute these values into the formula:


Solve a Quadratic Equation Using a Graph
To solve a quadratic equation using a graph, plot the quadratic equation on a graph and see where it crosses the x-axis.
Here is an example of a quadratic equation plotted on a graph. It is solved by x = 1 and x = 2:


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Read more about how to solve quadratic equations
Read more about how to solve quadratic equations using factoring
Read more about how to solve quadratic equations using factoring (where a ≠ 1)
Read more about how to solve quadratic equations using the quadratic formula
Read more about how to solve quadratic equations using a graph

Difference of Squares

A difference of squares is one squared quantity (a number or letter multiplied by itself) subtracted from another squared quantity. Some quadratic equations are in the form of a difference of squares. They can be factored and solved in an easy way. In this mini-curriculum, you will learn about the difference of squares.

Difference of Squares
A difference of squares is square number (a number multiplied by itself) subtracted from another square number.
Here is an example of a difference of squares:

One square number (22 = 2 × 2) is being subtracted from another square number (32 = 3 × 3).

Solve a Quadratic Equation Using a Difference of Squares
To solve a quadratic equation using a difference of squares, factor and solve the quadratic equation.

(1) Factor the Equation
Write the equation as two brackets multiplying each other.
Here is an example of factoring a quadratic equation into two brackets multiplying each other using a difference of squares:

x2 − 9 = 0
⇒ (x + 3)(x − 3) = 0

(2) Solve the Equation
Equate each bracket to 0 to find the two values of x that solve the equation.
Here is an example of equating each bracket to solve the equation:

(x + 3)(x − 3) = 0
x + 3 = 0 ⇒ x = −3
x − 3 = 0 ⇒ x = 3

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Read more about a difference of squares
Read more about how to solve quadratic equations using a difference of squares

Completing the Square

Completing the square is a way of writing a quadratic equation. In this mini-curriculum, you will learn about completing the square.

Perfect Square Trinomials
A perfect square trinomial is the result of squaring a binomial (two terms added or subtracted together).
Here is an example of a perfect square trinomial:


Completing the Square
Completing the square uses a perfect square trinomial to rewrite quadratic equations.
It is called completing the square because the method can be visualised as a square.


Complete the Square
To complete the square on a quadratic equation, write it as a squared binomial plus (or minus) a number.
Here is an example of completing the square on a quadratic equation:


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Read more about perfect square trinomials
Read more about completing the square
Read more about how to complete the square