Quadratic Equations

A quadratic equation is an equation in the form:

• x is a variable. It is an unknown.
• a, b and c are constants, where a cannot equal 0. They stand in for numbers.

Understanding Quadratic Equations

It is easier to understand quadratic examples with an example. Let's look at a quadratic equation. This is the quadratic equation shown at the start of the lesson with a = 2, b = 3 and c = 4. This a quadratic equation because of the x². A quadratic equation is an equation where the highest power of x is 2.

What Quadratic Equations Are...

Here are some real examples of quadratic equations. They are all in the standard form ax² + bx + c = 0:

3x2 + 5x + 2 = 0	a = 3, b = 5, c = 2
x2 + 4x + 3 = 0	a = 1, b = 4, c = 3 (no need to write 1x2)
x2 − 4x + 4 = 0	a = 1, b = −4, c = 4

The following equations are also quadratic equations, but they are not in standard form:

x2 = 2 − 5x	This looks different because some terms have been moved to the other side of the equals sign.
4x2 + 1 = 0	There is no x term (so b = 0), but the highest power of x is still 2.
2(x2 − x) = 3	The left hand side is factored and another term has been moved to the other side of the equals sign.
x(x + 2) = 0	When the bracket is expanded, you get a x2 term.

...And What Quadratic Equations Aren't

These equations are not quadratic equations:

x + 2 = 0	There is no x2 term (a = 0). This is a linear equation.
x3 + 2x2 + 3x + 4 = 0	There is a x3 term. The highest power of x is 3, not 2. This is a cubic equation.

The Graph of a Quadratic Equation

A quadratic equation can be plotted on a graph. Plot the function: The graph of a quadratic equation looks like a curve:

• y and x are the Cartesian coordinates of the points on the curve.

What's in a Name?

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

Interactive Widget

You can use this interactive widget to create a graph of your quadratic equation. Use the buttons to change the values of the quadratic equation.

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