Quadratic equations are really important for solving problems where we need to find the biggest or smallest values. These equations look like this:

$$y = ax^2 + bx + c$$

In this formula, $a$, $b$, and $c$ are constants, and $a$ can’t be zero. Let's take a look at some areas where we can use quadratic equations to solve these kinds of problems.

1. Maximizing Area

A common problem is figuring out how to make a rectangle with the biggest area when we know the outside distance (perimeter).

For example, if the perimeter of a rectangle is 40 meters, we can use a quadratic equation to find the best width ($w$) and length ($l$). The equation relating the two is:

$$l + w = 20$$

To find the area, we can replace $w$ with $(20 - l)$ in the area formula:

$$A = l \times w$$

So now we have:

$$A = l(20 - l) = 20l - l^2$$

This becomes a quadratic equation in the form of:

$$A = -l^2 + 20l$$

Here, $a$ is -1 and $b$ is 20. To find the biggest area, we use the vertex formula:

$$l = -\frac{b}{2a}$$

For this example:

$$l = -\frac{20}{2 \cdot -1} = 10 \text{ meters}$$

2. Projectile Motion

We can also use quadratic equations to understand how objects move when they are thrown into the air, which helps us find the highest point they reach and how far they go.

The height $h$ of a thrown object can be shown as a quadratic equation related to time $t$. For example:

$$h = -5t^2 + 20t + 1$$

This equation shows how high the object is at different times. The $-5$ shows how gravity pulls it down. To find out when it reaches the maximum height, we can use the vertex formula again:

$$t = -\frac{b}{2a} = -\frac{20}{2 \cdot -5} = 2 \text{ seconds}$$

We can then plug $t = 2$ back into the height equation to find the highest point.

3. Revenue Maximization

Businesses also use quadratic equations to find out the best price to charge so they can make the most money.

For example, if the revenue $R$ from selling $x$ items is shown as:

$$R = -2x^2 + 120x$$

The maximum revenue happens when:

$$x = -\frac{b}{2a} = -\frac{120}{2 \cdot -2} = 30 \text{ items}$$

This helps businesses know the best number of items to produce.

Conclusion

Quadratic equations are super useful in many real-life situations, like geometry, physics, and business. By using the vertex formula and understanding quadratic functions, we can easily solve problems and make better decisions.