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
y
x2
x
y-intercept

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?