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

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.

Practice