Quadratic Equations
(KS4, Year 10)
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 ax2 + bx + c = 0, where a, b, and c are constants and x is unknown."
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:
What's in a Name?
The word "quadratic" comes from the word "quad", meaning "square" - because the
x is squared.
