Quadratic Equations
(KS4, Year 10)
The Lesson
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.
The Oxford English Dictionary defines a quadratic equation as "an equation of the second degree; specifically an equation of the form ax^{2} + 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.
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
x^{2}.
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
ax^{2} + bx + c = 0:
3x^{2} + 5x + 2 = 0 
a = 3, b = 5, c = 2 
x^{2} + 4x + 3 = 0 
a = 1, b = 4, c = 3 (no need to write 1x^{2}) 
x^{2} − 4x + 4 = 0 
a = 1, b = −4, c = 4 
The following equations are also quadratic equations, but they are not in standard form:
x^{2} = 2 − 5x 
This looks different because some terms have been moved to the other side of the equals sign. 
4x^{2} + 1 = 0 
There is no x term (so b = 0), but the highest power of x is still 2. 
2(x^{2} − 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 x^{2} term. 
...And What Quadratic Equations Aren't
These equations are
not quadratic equations:
x + 2 = 0 
There is no x^{2} term (a = 0). This is a linear equation. 
x^{3} + 2x^{2} + 3x + 4 = 0 
There is a x^{3} 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:
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?