Examples of Word Problems Using Quadratic Equations in Real Life

Quadratic equations can be found in many everyday situations. They look like this: $ax^2 + bx + c = 0$. These equations often show relationships that involve squares. You can find them in things like physics, money matters, and even gardening! Let’s look at some easy-to-understand problems that can be solved using quadratic equations.

1. Throwing a Ball

One common example is when you throw a ball into the air. Imagine you’re on a bridge and you toss a ball straight up. The height of the ball $h$ (in meters) can be described based on how much time $t$ (in seconds) has passed with this equation:

$h(t) = -4.9t^2 + v_0 t + h_0$

In this equation, $v_0$ is how fast you throw the ball, and $h_0$ is how high you are when you throw it.

Example Problem: What if you throw a ball up with a speed of $20 \, \text{m/s}$ from 5 meters above the ground? When will it hit the ground?

Solution: We can set $h(t) = 0$ for when it hits the ground:

$0 = -4.9t^2 + 20t + 5$

Now we have a quadratic equation to solve for $t$.

2. Finding Area

Another cool use of quadratics is in area problems—especially when planning gardens or rectangular fields.

Example Problem: Let’s say you want to make a rectangular garden with an area of $100 \, \text{m}^2$. If the length is 5 meters longer than the width, what will the measurements be?

Solution: Let’s call the width $w$. The length can be written as $l = w + 5$. The area can be calculated like this:

$A = l \times w = (w + 5) \times w = 100$

This leads us to the quadratic equation:

$w^2 + 5w - 100 = 0$

By solving this equation, we can find how wide the garden is and then figure out the length.

3. Making Money

Quadratic equations are also useful in business when we want to make the most profit.

Example Problem: Imagine you run a small shop, and your profit $P$ (in hundreds of dollars) can be described by this equation:

$P(x) = -2x^2 + 12x + 5$

Here, $x$ is the number of items sold (in hundreds). How many items do you need to sell to get the highest profit?

Solution: To find the highest profit, we can use a special formula to find the peak:

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

In our equation, $a = -2$ and $b = 12$:

$x = -\frac{12}{2 \times -2} = 3$

This means selling 300 items (since $x$ is in hundreds) will give you the best profit!

4. Distance of a Moving Car

In physics, we often study how objects move. For example, if we know how far a car travels, we can use a quadratic equation:

Example Problem: The distance $d$ (in meters) that a car travels, with an acceleration of $2 \, \text{m/s}^2$, can be expressed as:

$d = 2t^2 + 5t$

How far does the car go in 3 seconds?

Solution: Plugging $t = 3$ into the equation gives:

$d = 2(3^2) + 5(3) = 18 + 15 = 33 \, \text{meters}$

Conclusion

As we’ve shown, quadratic equations are really useful and can help us solve different real-world problems, like motion, area, and making money. Learning how to turn these word problems into quadratic equations is an important skill, especially in middle school math. So next time you face a question about area, motion, or profit, think about how a quadratic equation might be the way to solve it!