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Imagine setting the ticket price for a music festival: set it too low, and you pack the venue but make no profit; set it too high, and the arena sits empty.
Finding the sweet spot where your total profit stays above a target threshold is a classic problem solved by quadratic inequalities.

The Quadratic Optimization Curve

A quadratic function is any relation of the form f(x)=ax2+bx+cf(x) = ax^2 + bx + c, which graphs as a symmetrical curve called a parabola.
Because parabolas are symmetric across their center, the maximum or minimum value occurs at the vertex, located at x=βˆ’b2ax = -\frac{b}{2a}.
To find where a quadratic quantity meets or exceeds a target value kk, we set up ax2+bx+cβ‰₯kax^2 + bx + c \ge k and rewrite it in standard form as Ax2+Bx+Cβ‰₯0Ax^2 + Bx + C \ge 0.
The boundary values where Ax2+Bx+C=0Ax^2 + Bx + C = 0 are called critical values, dividing the real number line into test intervals.
How do we apply this to business pricing when customer turnout drops as prices rise?

Scenario 1: Revenue and Profit Modeling

A community theater charges \20perticketandsellsanaverageofper ticket and sells an average of300$ seats per show.
Market research shows that for every \2increaseinticketprice,increase in ticket price,15$ fewer people attend.
Let xx be the number of \2priceincreases.Thenewpriceisprice increases. The new price is(20 + 2x)andthenumberofticketssoldisand the number of tickets sold is(300 - 15x)$.