Applications of Quadratics

A quadratic equation is a polynomial equation, which means the highest degree of the variable is 2. Its standard form of the quadratic equation is \(ax^2 + bx + c = 0\), where \(a ≠ 0\). 

Quadratic equations are used in everyday life to solve problems related to motion, area, finance, and design. In this section, we will learn how quadratic equations are used. 

Projectile motion: When a stone skips across a pond, it follows a curved path that is known as projectile motion, which can be described using quadratic equations. By factoring in the initial speed and launch angle, engineers and scientists can predict the following:
 

• The quadratic equation can be used to determine the maximum height a projectile reaches, which is essential for events like fireworks' success. They are also used to calculate the maximum range of a firefighter’s water cannon.
   
• In sports like the javelin throw, golf, and archery, the total distance traveled is calculated using a quadratic equation. 
   
• In a defense system, the quadratic equation is used to predict the landing point of projectiles like missiles by modeling their curved trajectories under the influence of gravity.

In business, quadratic equations are used to identify the point at which profits are maximized. By representing the relationship between profit and the number of units sold in quadratic form. By analyzing this relationship through a quadratic equation, companies can determine: 

The optimal number of units to produce or sell to maximize profit

The most effective pricing strategy that balances production costs and consumer demand to achieve the highest possible revenue

To predict the height of a falling object, use quadratics to model motion under gravity. 
 

Designing: In fields like architecture, engineering, and technology, quadratic equations are used to create precise and effective designs. 

Satellite dishes and radio telescopes: The parabolic design of satellite dishes and large telescopes is derived from the quadratic equations. This shape allows incoming signals, whether from satellites or distant galaxies, to a single receiver point, resulting in clearer images and clearer sound transmissions. 

Civil engineering and structural design: In civil engineering, quadratic equations are used in the design and analysis of bridges, ramps, arches, and dams. It helps to ensure safety, functionality, and efficiency in infrastructure projects. 
 

• Quadratic equations are used to model the curved and parabolic shapes in bridges and arches, which helps engineers to accurately calculate load-bearing capacity and to optimize stress distribution. 
   
• To model the stable tunnel curves, engineers apply quadratic models to reduce stress on surrounding materials and improve long-term safety. 
   
• When constructing the dams, engineers use quadratic principles to evenly distribute water pressure and prevent structural failure. 

Mastering the value of a polynomial means getting perfect at substituting numbers into expressions and simplifying them correctly. Here are some tips and tricks to master the value of a polynomial.
 

1. Think of It as a number machine. Say to yourself that, “A polynomial is like a machine — I put in a number for 𝑥, and it gives me an output.” This helps visualize what’s happening when you substitute.
    

2. Always use brackets when substituting. This avoids sign mistakes, one of the most common errors.
    

3. Practice with different values for x. Like, x=0 (checks the constant term) x=1 (easy for mental math). If your answers make sense for all types of values, you’ve mastered the skill.
    

4. Double-check with a calculator. After doing it by hand, use a calculator to confirm. Compare results to spot where you might’ve gone wrong, great for self-correction.
    
5. Check your work by substitution. Plug the value back into the polynomial step-by-step and see if it makes sense. For example, results should increase for bigger 𝑥 if the leading coefficient is positive.