Class 10th Maths Ex 4.4

Mastering Class 10th Maths Ex 4.4: A Comprehensive Guide to Quadratic Equations

This article provides a comprehensive guide to solving problems from Class 10th Maths Ex 4.4, focusing on quadratic equations. We will explore various methods for solving quadratic equations, delve into the underlying mathematical concepts, and address common challenges students face. By the end, you'll not only be able to solve problems from this exercise but also have a deeper understanding of quadratic equations themselves. This guide includes numerous examples, step-by-step solutions, and frequently asked questions to ensure a thorough understanding.

Introduction to Quadratic Equations

A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable (usually denoted as 'x') is 2. The general form of a quadratic equation is:

ax² + bx + c = 0

where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero (a ≠ 0). If a = 0, the equation becomes linear, not quadratic. Solving a quadratic equation means finding the values of 'x' that satisfy the equation—these values are called the roots or solutions of the equation.

Methods for Solving Quadratic Equations

Exercise 4.4 typically covers several methods for solving quadratic equations. Let's explore the most common ones:

1. Factorization:

This method involves expressing the quadratic expression as a product of two linear factors. Once factored, we can set each factor equal to zero and solve for 'x'. This method is particularly useful when the quadratic expression can be easily factored.

Example:

Solve the quadratic equation: x² + 5x + 6 = 0

Solution:

1. Factor the quadratic expression: (x + 2)(x + 3) = 0
2. Set each factor equal to zero: x + 2 = 0 or x + 3 = 0
3. Solve for x: x = -2 or x = -3

Therefore, the roots of the equation are x = -2 and x = -3.

2. Completing the Square:

This method involves manipulating the quadratic equation to create a perfect square trinomial on one side of the equation. This allows us to easily solve for 'x' by taking the square root of both sides. This method is particularly useful when factorization is not straightforward.

Example:

Solve the quadratic equation: x² - 6x + 5 = 0

Solution:

1. Move the constant term to the right side: x² - 6x = -5
2. Complete the square: To complete the square for x² - 6x, we take half of the coefficient of x (-6/2 = -3) and square it (-3)² = 9. Add 9 to both sides: x² - 6x + 9 = -5 + 9
3. Simplify: (x - 3)² = 4
4. Take the square root of both sides: x - 3 = ±2
5. Solve for x: x = 3 ± 2 Therefore, x = 5 or x = 1

3. Quadratic Formula:

The quadratic formula is a general method that can be used to solve any quadratic equation, regardless of whether it can be easily factored. The formula is derived from completing the square and is given by:

x = [-b ± √(b² - 4ac)] / 2a

where 'a', 'b', and 'c' are the coefficients of the quadratic equation ax² + bx + c = 0.

Example:

Solve the quadratic equation: 2x² - 5x + 2 = 0

Solution:

Here, a = 2, b = -5, and c = 2. Substitute these values into the quadratic formula:

x = [5 ± √((-5)² - 4 * 2 * 2)] / (2 * 2) x = [5 ± √(25 - 16)] / 4 x = [5 ± √9] / 4 x = [5 ± 3] / 4

Therefore, x = 2 or x = 1/2

Understanding the Discriminant (b² - 4ac)

The expression b² - 4ac within the quadratic formula is called the discriminant. The discriminant tells us about the nature of the roots of the quadratic equation:

• b² - 4ac > 0: The equation has two distinct real roots.
• b² - 4ac = 0: The equation has one real root (a repeated root).
• b² - 4ac < 0: The equation has no real roots (the roots are complex numbers).

Solving Word Problems Involving Quadratic Equations

Many real-world problems can be modeled using quadratic equations. Exercise 4.4 likely includes word problems requiring you to translate the problem into a quadratic equation and then solve it using one of the methods described above. These problems often involve concepts like area, speed, and distance. Carefully analyze the problem to define the variables and formulate the appropriate equation.

Example (Area Problem):

A rectangular garden has a length that is 3 meters more than its width. If the area of the garden is 70 square meters, find the dimensions of the garden.

Solution:

Let the width of the garden be 'w' meters. Then the length is (w + 3) meters. The area is given by:

w(w + 3) = 70

Expanding and rearranging, we get a quadratic equation:

w² + 3w - 70 = 0

This equation can be solved by factorization, completing the square, or the quadratic formula. Solving this equation yields w = 7 (or w = -10, which is not physically possible since width cannot be negative). Therefore, the width is 7 meters and the length is 10 meters (7 + 3).

Step-by-Step Approach to Solving Problems from Ex 4.4

1. Understand the problem: Carefully read the problem statement and identify what is being asked.
2. Identify the variables: Define the variables involved in the problem.
3. Formulate the equation: Translate the problem into a quadratic equation.
4. Choose a method: Select the most appropriate method for solving the quadratic equation (factorization, completing the square, or the quadratic formula).
5. Solve the equation: Apply the chosen method to solve for the variable(s).
6. Check your solution: Verify that your solution satisfies the original problem statement.
7. Write the answer: Clearly state your answer in the context of the problem.

Frequently Asked Questions (FAQ)

• Q: What if I can't factor the quadratic equation easily? A: Use the quadratic formula. It works for all quadratic equations.

• Q: How do I know which method to use? A: Factorization is easiest when it's readily apparent. Completing the square is useful when factorization is difficult. The quadratic formula works always.

• Q: What if the discriminant is negative? A: This indicates there are no real solutions to the quadratic equation. The solutions are complex numbers, involving the imaginary unit 'i' (√-1).

• Q: How do I handle word problems? A: Carefully define variables, translate the problem into an equation, solve the equation, and then interpret the solution in the context of the problem.

Conclusion

Mastering Class 10th Maths Ex 4.4 requires a solid understanding of quadratic equations and the various methods for solving them. This article provides a thorough overview of these methods, along with examples and a step-by-step approach to tackling problems. Remember to practice regularly and work through various examples to build your confidence and proficiency. By understanding the concepts and applying the methods consistently, you can successfully solve any problem from Ex 4.4 and build a strong foundation in algebra. Don't hesitate to review the material and practice until you feel comfortable with the process. Good luck!