Quadratic Equations
(KS4, Year 10)

The Lesson

A quadratic equation is an equation in the form:

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

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."

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.
Change the Equation

Understanding Quadratic Equations

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

2 x squared plus 3 x plus 4 equals 0 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:

y equals a x squared plus b x plus c The graph of a quadratic equation looks like a curve:

quadratic equation on a graph

What's in a Name?

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

What is an equation? What is a variable? What is a constant? What is a power? What is a square number? What are Cartesian coordinates?