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Welcome to our Math lesson on What are Quadratic Inequalities?, this is the first lesson of our suite of math lessons covering the topic of Quadratic Inequalities, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.
In the previous chapter, we dealt extensively with quadratic equations, which are equations with one variable raised to the second power. Recall that the general formula of quadratic equations with one variable is
where x is the variable, a and b are coefficients (numbers preceding a variable) and c is a constant (numbers not followed by a variable).
Obviously, linear inequalities must be similar to linear equalities except for the sign, which must be one of the four inequality signs explained in the previous tutorial. Hence, we obtain four possible general forms of quadratic inequalities:
For example,
is an example of a quadratic inequality, as it contains a single variable raised to the second power at maximum.
If the quadratic inequality is not in one of the four standard forms written above, we must turn them into the standard form, where the most appropriate thing would be to have the coefficient a positive. For example, if the original quadratic inequality is
we turn it into the standard form by applying the properties of inequalities explained in the previous tutorial. Thus,
Rearranging the terms from the highest to the lowest power yields
We said before that the coefficient a must be positive. Therefore, we have to multiply both sides by -1. During this procedure, we must not forget to swap the direction of the inequality sign. Thus, we obtain
Arrange the following quadratic inequalities to express them in the standard form.
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