What is the Quadratic Expression?
A quadratic expression is a statement in algebra that has a variable that is raised to the power of 2.
Please scroll to the bottom of the page for free practice exercises.
Quadratic Expression, Equation and Inequality
A quadratic expression should not be confused with a quadratic equation.
Quadratic equations contain an equals sign, while quadratic expressions do not.
Quadratic inequalities include the less than or greater than signs, while quadratic expressions do not.
ax2 + bx + c = 0 is an example of a quadratic equation.
ax2 + bx + c > 0 is an example of a quadratic inequality.
But quadratic inequalities and equations do include quadratic expressions.
Everything to the left of the equals sign or greater than sign in our examples above is a quadratic expression.
So, ax2 + bx + c = 0 is a quadratic expression.
Expanding Quadratics
Quadratics that contain terms in parentheses can be multiplied and expanded in order to make a quadratic expression.
If there is one set of parenthesis, the expression can be multiplied by using the distributive property of multiplication as shown below:
x(x + 2) =
(x × x) + (x × 2) =
x2 + 2x
The above expression is a quadratic expression because it contains x2.
If you have terms in two sets of parentheses, you can multiply using the FOIL method.
In order to use the FOIL method, you have to multiply the terms together in this order: FIRST OUTSIDE INSIDE LAST
Look at this example:
(x + 1)(2x + 3) =
FIRST OUTSIDE INSIDE LAST
(x × 2x) + (x × 3) + (1 × 2x) + (1 × 3) =
2x2 + 3x + 2x + 3 =
2x2 + 5x + 3
We explain the FOIL method in more depth in another post.
If you are not confident about how to solve algebra problems with the FOIL method, be sure to view that post.
Now you should try the exercises in the next section.
Quadratic Expression – Exercises
Instructions: Use multiplication in the problems below to expand and re-write them as quadratic expressions. Your answers should contain a variable raised to the power of two.
1) x(x – 4)
2) 2x(4x + 6)
3) –3x(x – 5)
4) (x + 2)(x – 3)
5) (3x – 3)(2x – 4)
Quadratic Expression – Answers
1) x(x – 4) =
(x × x) + (x × –4) =
x2 – 4x
2) 2x(4x + 6) =
(2x × 4x) + (2x × 6) =
8x2 + 12x
3) –3x(x – 5) =
(–3x × x) + (–3x × –5) =
–3x2 + 15x
4) (x + 2)(x – 3) =
(x × x) + (x × –3) + (2 × x) + (2 × –3) =
x2 + –3x + 2x + –6 =
x2 – x – 6
5) (3x – 3)(2x – 4) =
(3x × 2x) + (3x × –4) + (–3 × 2x) + (–3 × –4) =
6x2 + –12x + –6x + 12 =
6x2 – 18x + 12
You might also like to have a look at our posts on the quadratic formula and quadratic equations.