📘 What is a Quadratic Equation?

A quadratic equation is a mathematical equation in which the highest power of the variable (x) is 2.

It is written in the standard form as:

[
ax^2 + bx + c = 0
]

where

• ( a, b, c ) are constants, and
• ( a ≠ 0 ).

🧮 Example:

[
2x^2 + 3x – 5 = 0
]

Here,

• ( a = 2 ),
• ( b = 3 ),
• ( c = -5 ).

✳️ Formula to Solve Quadratic Equations

You can find the value of ( x ) (called the roots) using the quadratic formula:

[
x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}
]

The part under the square root, ( b^2 – 4ac ), is called the discriminant — it tells you the nature of the roots (real, equal, or imaginary).

🌍 Where Are Quadratic Equations Used in Real Life?

Quadratic equations are not just math — they describe many real-world situations involving curves, motion, and optimization.

Here are some everyday examples 👇

1. 🎯 Projectile Motion (Throwing or Shooting Objects)

When you throw a ball, its path is curved (parabolic) — and this curve can be modeled using a quadratic equation.

Example:
A ball is thrown upward with an initial velocity.
The height ( h ) after time ( t ) is given by:

[
h = -4.9t^2 + 20t + 1
]

This equation helps find:

• The maximum height of the ball,
• The time it takes to hit the ground.

2. 🚗 Design and Engineering

Engineers use quadratic equations to:

• Design parabolic bridges or arches,
• Model headlight reflectors or satellite dishes, which are parabolic in shape.

Example:
The curve of a car’s headlight reflector is shaped using the equation:
[
y = ax^2
]
so that all reflected light rays focus at a single point.

3. 💰 Business and Economics

Quadratic equations help find maximum profit or minimum cost — a process called optimization.

Example:
A company’s profit ( P ) may depend on the number of products ( x ) sold:
[
P = -2x^2 + 40x – 100
]
Here, solving gives the number of units to sell for maximum profit.

4. 🎢 Sports and Amusement Rides

The shape of roller coaster tracks, ski jumps, and basketball arcs can be modeled using quadratic equations for safe and smooth motion.

5. 📐 Architecture

Architects use quadratics to design arches, domes, and curved roofs.
Example: The Gateway Arch in the USA follows a quadratic curve.

🧠 Summary Table