Our daily lives involve regular use of our mathematical knowledge to solve real-life problems. Just like other mathematical concepts, we also use quadratic equations unknowingly to find answers to our questions.

A quadratic equation is an equation containing variables, among which at least one must be squared. It is expressed in the following form:

ax^{2}+bx+c= 0

Here, ‘x’ is the unknown value we need to calculate. The letters ‘a’ and ‘b’ represent the known numbers you put in while calculating. However, one must remember that ‘a’ can never be zero.

Although simply looking at the equation you may feel that it is not something you use very often. So, let us give you a few examples of how quadratic equations find their application in our daily lives.

**Real-life applications of quadratic equations**

**1. Building a home? How do you calculate areas?**

Constructors and architects take the help of quadratic equations to develop a building. For any building, you need to calculate how much land you have, how big each room will be, what the shape of the building will be, etc. So, if you are building a rectangular house, you know one side of the house will be bigger than the other. So, you can use the ratio of these two sides to calculate the materials you need.

When you buy a piece of land, you know the total size of it in square feet. Your constructor or builder plans the whole house for you, dividing the total available area across your home. They do so by using the quadratic equation. The size of land, number of rooms, etc., are known factors (like a,b), while the size of rooms and space allocation for stairs or corridors are unknown (the ‘x’). Now, put these values to the equation ax2+bx+c= 0, and you have all the data you need.

**2. Running a business? How much profit are you making?**

To calculate the profit you earn or are likely to earn from your business, you need the quadratic equation’s help. Whatever product or service you sell, you know how much it costs to produce. You are also aware of the amount of profit you aim to make. So, you might wonder what your product should be priced at. This unknown selling price is your ‘x’ of the quadratic equation. The cost price and the profit are known factors, ‘a’ and ‘b.’ Now that you have all the elements of a quadratic equation solving it step by step will give you the selling price.

**3. How much speed do you need? Estimate with quadratic equations.**

Calculating the required speed calls for the use of a quadratic equation. You will find it even more helpful if you are on your college or university rowing team. If you are going upstream, you have to row against the stream. But you need to know how fast you should row or what should be the speed of your boat. You can easily recognize that by using the quadratic equation. Let’s first look at the known factors. You know the distance you have to row to and back, the speed of the stream, and the total time available to you. Assuming your speed is ‘x,’ you can put these known values in the ax^{2}+bx+c= 0 equation. Now continue to solve it step-by-step; you will have the answer ready in no time.

**4. Are you into sports? You need quadratic equations too!**

Velocity quadratic equations are something that athletes and sports analysts use every day, every moment, especially in the case of basketball, javelin throw, shot put, etc., which involves throwing balls or spears or other such items.

In a basketball team, you will see that one player throws the ball to another, reaching them in a moment after or before they catch it. It will feel like the player knows when he should throw the ball so that it reaches the other player at the exact time. It is really a matter of quadratic calculation. Here, the known values of ‘a’ and ‘b’ are the heights, speed of the ball, loss of speed due to gravitational force, etc. The time is unknown to you, and this is the ‘x.’ Thus, you can solve it by referring to the ax^{2}+bx+c= 0 equation. This is how your favorite basketball players score.

**5. Use of quadratic equations by engineers**

Among all the professions, engineers probably use quadratic equations most extensively. Here’s a list:

- Engineers in the automobile industry use quadratic equations to design vehicle structures, especially those with curved patterns.
- Automobile engineers use quadratic equations for designing and installing brake systems.
- Aerospace engineers use these equations for calculating a vertical plane projectile. Calculating the height and velocity of an object when thrown or launched in space, they use the ax
^{2}+bx+c= 0 formula to recognize how much time it will take to reach the destination. - Using complex systems at the workplace calls for regular and frequent use of quadratic equations by electrical and chemical engineers.
- Audio engineers also use quadratic equations to design sound systems to ensure the listeners experience the best sound quality.

**6. Satellite dish setting and signal transmission**

You need to put up a satellite dish at a particular angle to receive the most efficient signal transmission. The dish on your roof catches the signals from two or more satellites simultaneously and transmits them to your TV through a feed horn. This whole transmission process involves using quadratic equations to identify the most efficient angle.

**7. Use of quadratic equations in defense and military services**

We already told you how to use quadratic equations to measure height, distance, speed, etc. These measurements are also used in defense services and military activities. For example, if the military needs to throw artillery to destroy an enemy camp, they calculate the distance and the speed of launching the artillery through quadratic equations.

**8. Do you wish to have a farm? Even then, you need quadratic equations.**

At first, agriculture, farming, and quadratic equations might seem like two entirely different topics, but a successful agricultural venture involves quadratic equations. These equations are used to calculate the total available area and determine how the area will be divided and how crops should be allocated. Randomly building a pen anywhere on the farm will not help you achieve high yields. The use of quadratic equations ensures agricultural efficiency.

**9. Application of quadratic equations= Effective management**

In any industry, there are various levels of management. A Production Manager supervises the product line manufacturing and other relevant activities. Again, an Engineering Manager supervises the machinery, work efficiency, etc. To estimate all these, they take the help of quadratic equations. The Human Resource Managers determine the workforce capability, requirement of recruits, etc., by considering the available work as the known variable. Thus, management also applies quadratic equations quite unknowingly.

**10. Other real-life applications of quadratic equations**

In addition to the above examples, there are other real-life instances where quadratic equations are used. These include:

- Astronomers identify and describe solar systems, planets and their orbits, and galaxies with the help of quadratic equations.
- Computer engineers ease the use of complex systems through quadratic equation implementation.
- Criminal investigators determine the trajectories of bullets by using quadratic equations.
- Insurance agents design plans and models through the computation of data. These plans, in most cases, are unique to each consumer. Such a complex process requires in-depth quadratic equation calculation.
- If there is a car accident, quadratic equations help determine car speeds.
- In Physics, different motions are described easily with quadratic equations.
- Chemical Engineers or Chemists working in fields of Chemistry use these equations to describe specific chemical reactions, identify their equilibrium, etc.

**Wrapping up**

Quadratic equations help us evaluate the relationship between variable quantities. Right from calculating area, speed, and profits to astronomy and criminal investigations, this math concept is useful in every sphere of life.

The above examples clearly show how we use quadratic equations in our daily lives. Therefore, developing a thorough understanding of the subject is necessary if we don’t wish to get stuck finding answers to our everyday queries.