**Quadratic applications** are very helpful in solving several types of word problems, especially where optimization is involved. Again, we can use the vertex to find the maximum or the minimum values, and roots to find solutions to quadratics.

Note that we did a **Quadratic Inequality Application **problem here in the** Quadratic Inequalities **section. Note also that we will discuss **Optimization Problems using Calculus** in the **Optimization Using Calculus** section.

## Quadratic Projectile Problem:

**Quadratic Projectile** problems are common quadratic application problems. Here is an example:

(We will discuss projectile motion using **parametric equations** here in the **Parametric Equations** section.)

## Quadratics Trajectory (Path) Problem

Note that typically, the independent variable represents** time, **not ** distance; however, **sometimes parabolas represent the distance on the $ x$-axis and the height on the $ y$-axis, and the shapes are similar.

In these problems, the $ x$-axis is measuring the horizontal distance of the path of the ball, not the time, so the parabola is a true indication of the **trajectory** or **path** of the ball.

## Optimization of Area Problem:

A common application of quadratics is **optimization**, which typically involves finding the vertex of a parabola since it’s the highest or lowest amount. Here’s a problem:

## Maximum Profit Problem:

Here’s another **optimization** problem:

## Maximum Revenue Problem:

Here’s another **optimization** problem:

## Bunny Rabbit Population Problem:

## Linear Increase/Decrease Problem:

OK, use your imaginations on this one 🙂

## Pythagorean Theorem Quadratic Application:

Here’s one that uses the **Pythagorean Theorem**:

## Quadratic Inequality Problem:

You may encounter a **Quadratic Inequality** problem like this:

## Finding Quadratic Equations from Points or a Graph

We saw how to obtain a Quadratic Equation from a point and/or graph here in the **Solving Quadratics by Factoring and Completing the Square** section. Here is a problem addressing this:

**Solution:**

**Learn these rules, practice, practice, practice, and you’ll rock at math!**

On to **Imaginary (Non-Real) and Complex Numbers**** **– you’re ready!