A quadratic inequality is solved by finding where the parabola lies above or below the -axis. The same ideas, together with the vertex, answer maximum problems about revenue, area and height.
Quadratic inequalities
Example 9.7.1. Solve .
Show solution
X: top , bottom , sides and . So
Example 9.7.2. Solve .
Show solution
X: top , bottom , sides and . So
The parabola dips below the axis between and , so it is positive outside them.
Example 9.7.3. Solve .
Show solution
First solve the equation:
Testing each region of the number line: false to the left of , true between and , false to the right of . So
Example 9.7.4. Solve .
Show solution
:no solution.
Rearranged, the inequality is , and
using the factor tree , so .
There are no real roots and the parabola opens up (it is sketched as a “U”), so it is never below .
Completing the square gives a second way to solve an inequality: it turns the quadratic into a bound on an absolute value, as the first two exercises at the end of this page do.
Applications
Revenue
Revenue:, where is the number of items sold and is the price.
Example 9.7.5. The price is . Find the maximum revenue and the price for maximum revenue.
Show solution
For instance, dollars.
Domain. We need :
which is zero at and at . The parabola opens down, so it is non-negative between these:
This means we do not want to produce more than items or less than items, to have positive revenue.
Max revenue. The vertex is with :
so the vertex is .
The price at max revenue is dollars.
Area
Example 9.7.6. Fence for a rectangular area: find the max area available with ft of perimeter.
Show solution
Constraint:
This is the limiting / constraint equation.
Area: is the maximizing equation. Substituting the constraint,
The vertex is at
where
Thus the vertex is : the maximum area is ft² at .
Height
Example 9.7.7..
Show solution
Setting
gives where it lands. The coefficient on gives the max height, at the vertex
Exercises
Exercise 9.7.8. Find all such that .
Show solution
. Completing the square, , so
Taking square roots, , so :
(Equivalently, gives : and .)
Exercise 9.7.9. Find all such that .
Show solution
so .
Exercise 9.7.10.Show solution
Price (value at a given :); tickets per game; each one-dollar increase loses 100 tickets, so the tickets sold are
which is at . At the revenue is . We want