Problem

Solve the equation 4x - 7 = 9 to find the value of x.

Key Equations

Solution

To solve for x, first add 7 to both sides: 4x - 7 + 7 = 9 + 7, resulting in 4x = 16. Next, divide both sides by 4: x = 16 ÷ 4, which simplifies to x = 4.

Explanation

This method works because we are isolating the variable x by performing inverse operations. Adding 7 cancels out the -7, and dividing by 4 gives us the value of x directly.

Problem

Calculate the volume of a cylinder with a radius of 3 cm and a height of 5 cm using the formula V = πr²h.

Key Equations

Solution

First, substitute the values into the formula: V = π × (3 cm)² × 5 cm. Calculate the area of the base: (3 cm)² = 9 cm². Then, multiply by the height: V = π × 9 cm² × 5 cm = 45π cm³. Thus, the volume is approximately 141.37 cm³ when using π ≈ 3.14.

Explanation

The volume formula for a cylinder derives from the area of the circular base multiplied by the height. This method allows us to find how much space the cylinder occupies in three dimensions.

Problem

Find the roots of the quadratic equation 2x² - 4x - 6 = 0 using the quadratic formula.

Key Equations

Solution

First, identify coefficients: a = 2, b = -4, c = -6. Apply the quadratic formula: x = (-b ± √(b² - 4ac)) / 2a. Calculate the discriminant: b² - 4ac = (-4)² - 4 × 2 × (-6) = 16 + 48 = 64. Now substitute: x = (4 ± √64) / 4. Since √64 = 8, we get two solutions: x = (4 + 8) / 4 = 3 and x = (4 - 8) / 4 = -1.

Explanation

The quadratic formula provides a systematic way to find the roots of any quadratic equation. By calculating the discriminant, we determine the nature of the roots, whether real and distinct, real and repeated, or complex.

Code

def factorial(n):  # Define a function to calculate factorial
    if n == 0 or n == 1:  # Base case for recursion
        return 1
    else:
        return n * factorial(n - 1)  # Recursive call

# Usage
result = factorial(5)  # Calculate factorial of 5
print(f'The factorial of 5 is {result}')

Explanation

This code defines a recursive function to calculate the factorial of a number. The function checks if the input is 0 or 1, returning 1 in those cases. For other numbers, it calls itself with a decremented value until it reaches the base case. This illustrates recursion, a fundamental programming concept where a function calls itself to solve smaller instances of the same problem.

Use Case

Calculating factorials is common in combinatorial problems, such as determining permutations and combinations in statistics or probability calculations.

Output

The factorial of 5 is 120

Code

numbers = [1, 2, 3, 4, 5]  # List of numbers
squared_numbers = [n ** 2 for n in numbers]  # List comprehension to square each number

# Usage
print(f'Squared numbers: {squared_numbers}')  # Print the squared numbers

Explanation

This code snippet uses list comprehension, a concise way to create lists in Python. It iterates over a predefined list of numbers, squaring each number and storing the results in a new list. This showcases Python's ability to handle data transformations efficiently and elegantly, reducing the need for verbose loops.

Use Case

List comprehensions are widely used in data processing tasks, such as transforming datasets or preparing data for machine learning algorithms, where efficiency and readability are crucial.

Output

Squared numbers: [1, 4, 9, 16, 25]