Identifying Quadratic Equations
For this web exercise, you need to fully understand what terms are. So, you may want to check these out:
- Recognizing Products and Sums; Identifying Factors and Terms
- Identifying Variable Parts and Coefficients of Terms
DEFINITION
quadratic equation
Let $\,a\,,$ $\,b\,$ and $\,c\,$ be real numbers,
with $\,a\ne 0\,.$
A quadratic equation is an equation of the form:
$$ax^2 + bx + c = 0$$
Important notes about the definition:
- A quadratic equation is, first and foremost, an equation. It must have an ‘$\,=\,$’ sign.
- When mathematicians say ‘an equation of the form ...’ they really mean ‘an equation that can be put in the form ...’ by using the two primary equation-solving tools: the Addition Property of Equality and the Multiplication Property of Equality.
- A quadratic equation must have an $\,x^2\,$ term. This is what $\,a\ne 0\,$ tells us.
- A quadratic equation is allowed (but not required) to have an $\,x\,$ term. The coefficient $\,b\,$ might be zero, which means the $\,x\,$ term is gone.
- A quadratic equation is allowed (but not required) to have a constant term. (Recall that a constant term is just a number—no variables.) The constant term, $\,c\,,$ might be zero.
So, to check if an equation is a quadratic equation, you want to make two passes through it (both sides):
- Does it have an $\,x^2\,$ term appearing somewhere? If not, then it's not a quadratic equation. Note: it can have lots of $\,x^2\,$ terms!
- The only other two term types that are allowed are $\,x\,$ terms and constants terms. (For example: no $\,x^3\,$ terms, no variables inside square roots, no variables in denominators, and so on.) So, sweep across the equation and look for anything other than $\,x\,$ terms and constant terms. If you find any, then it's not a quadratic equation.
Examples
In this exercise, you will practice identifying quadratic equations.
Question:
Is
$\,x^2 = x + 4\,$ a quadratic equation?
Solution:
Does it have an $\,x^2\,$ term?
Check!
Anything other than $\,x\,$ terms or constant terms?
Nope.
Check!
YES, it is a quadratic equation.
Question:
Is
$\,3x - 4 = x + 1\,$ a quadratic equation?
Solution:
Does it have an $\,x^2\,$ term?
Nope.
So, it's not a quadratic equation.
Question:
Is
$\,x - 2x^2 = 1 + x^5\,$ a quadratic equation?
Solution:
Does it have an $\,x^2\,$ term?
Check!
Anything other than $\,x\,$ terms or constant terms?
Oops.
Quadratic equations are not allowed to have an $\,x^5\,$ term.
So, it's not a quadratic equation.