What is Modelling with Quadratic Equations?

S3-SA1-0462

Grade Level:

Class 8

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

Modelling with quadratic equations means using mathematical equations where the highest power of the variable is 2 (like x^2) to describe and predict real-world situations. These equations help us understand how things change in a curved or parabolic way, not just in a straight line.

Simple Example

Quick Example

Imagine a cricket player hitting a six! The path the ball takes through the air is not a straight line; it goes up and then comes down. We can use a quadratic equation to model this path, helping us predict how high the ball will go and where it will land.

Worked Example

Step-by-Step

Let's say a local shopkeeper wants to find the best price for a samosa to maximize his daily profit. He observes that if he sells samosas at Rs. 'x' each, his daily profit 'P' can be modelled by the equation: P = -5x^2 + 60x - 100. Let's find the profit if he sells samosas at Rs. 5 each.

Step 1: Write down the given quadratic equation: P = -5x^2 + 60x - 100
---
Step 2: Identify the value of 'x' we need to use. Here, x = 5 (price per samosa).
---
Step 3: Substitute x = 5 into the equation: P = -5(5)^2 + 60(5) - 100
---
Step 4: Calculate the square of x: P = -5(25) + 60(5) - 100
---
Step 5: Perform the multiplications: P = -125 + 300 - 100
---
Step 6: Perform the additions and subtractions: P = 175 - 100
---
Step 7: Calculate the final profit: P = 75

Answer: If the shopkeeper sells samosas at Rs. 5 each, his daily profit will be Rs. 75.

Why It Matters

Modelling with quadratic equations is super important in fields like AI/ML and Data Science to predict trends or optimize processes. Engineers use it to design bridges and buildings, while physicists apply it to understand projectile motion. It's a foundational skill for many future careers!

Common Mistakes

MISTAKE: Forgetting to square the variable 'x' before multiplying by its coefficient (e.g., calculating -5 * 5 instead of -5 * 5^2). | CORRECTION: Always follow the order of operations (BODMAS/PEMDAS). Exponents (powers) come before multiplication.

MISTAKE: Making sign errors, especially with negative numbers. Forgetting that a negative number squared becomes positive (e.g., (-2)^2 = -4). | CORRECTION: Remember that (-a)^2 = a^2. Pay close attention to negative signs during multiplication and subtraction.

MISTAKE: Confusing the 'x' in the equation with a constant. | CORRECTION: 'x' represents a variable quantity that can change, while other numbers in the equation (like -5, 60, -100) are constants that stay fixed for that specific model.

Practice Questions

Try It Yourself

QUESTION: The height 'h' (in meters) of a ball thrown upwards after 't' seconds is given by h = -5t^2 + 20t. What is the height of the ball after 3 seconds? | ANSWER: 15 meters

QUESTION: A farmer wants to fence a rectangular field. He has 100 meters of fencing. If one side of the field is 'x' meters, the area 'A' (in square meters) can be modelled by A = x(50 - x). Calculate the area if one side is 10 meters. | ANSWER: 400 square meters

QUESTION: The number of daily commuters 'N' (in thousands) using a new metro line, 'x' days after its inauguration, is given by N = -0.1x^2 + 5x + 10. How many commuters (in thousands) will use the metro line 10 days after inauguration? What will be the change in commuters from day 5 to day 10? | ANSWER: N(10) = 50 thousand commuters; Change = 17.5 thousand commuters

MCQ

Quick Quiz

Which of the following real-world situations is most likely to be modelled by a quadratic equation?

The cost of buying 'x' number of pens at Rs. 10 each.

The path of a stone thrown into a pond.

The amount of simple interest earned on a fixed deposit over 'x' years.

The total distance covered by a car moving at a constant speed for 'x' hours.

The Correct Answer Is:

Option B, the path of a stone thrown, follows a parabolic (curved) trajectory which is best described by a quadratic equation. Options A, C, and D represent linear relationships, not quadratic.

Real World Connection

In the Real World

From ISRO scientists calculating satellite trajectories to engineers designing the curved support structures for flyovers, quadratic equations are everywhere. Even app developers use them in gaming to simulate object movement, or in financial apps to predict stock price fluctuations over time, although these are often more complex models.

Key Vocabulary

Key Terms

Quadratic Equation: An equation where the highest power of the variable is 2, like ax^2 + bx + c = 0. | Model: To represent a real-world situation or system using mathematical equations. | Variable: A symbol (like x, t, h) that represents a quantity that can change. | Coefficient: A number multiplied by a variable in an algebraic term (e.g., 'a' in ax^2). | Parabola: The U-shaped curve that is the graph of a quadratic equation.

What's Next

What to Learn Next

Great job understanding modelling with quadratic equations! Next, you can explore 'Solving Quadratic Equations' to find specific values of 'x' that satisfy a given condition, or 'Graphing Quadratic Equations' to visually understand their parabolic shapes. These topics will deepen your understanding and problem-solving skills.