Exercise 3.4 Class 10 Maths

Exercise 3.4 Class 10 Maths: A Comprehensive Guide to Quadratic Equations

This article provides a thorough walkthrough of Exercise 3.4 from Class 10 mathematics textbooks, focusing on solving quadratic equations using the quadratic formula. We'll break down each problem, explain the underlying concepts, and offer helpful tips to master this crucial chapter. Understanding quadratic equations is fundamental for further mathematical studies, and this guide aims to equip you with the necessary tools and confidence to excel. We'll cover various problem types, demonstrate the application of the quadratic formula, and address common student challenges.

Introduction to Quadratic Equations and the Quadratic Formula

A quadratic equation is a polynomial equation of degree two, meaning the highest power of the variable (usually 'x') is 2. It generally takes the form: ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero (if a=0, it becomes a linear equation).

Solving a quadratic equation means finding the values of 'x' that satisfy the equation. There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. Exercise 3.4 typically focuses on the application of the quadratic formula, which is a powerful and universally applicable method:

x = [-b ± √(b² - 4ac)] / 2a

This formula provides the two possible solutions (roots) for 'x' in the quadratic equation. The term b² - 4ac is known as the discriminant, and it determines the nature of the roots:

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

Step-by-Step Guide to Solving Problems from Exercise 3.4

Let's now walk through the process of solving typical problems found in Exercise 3.4, using the quadratic formula. Remember, each problem will require identifying the values of 'a', 'b', and 'c' from the given quadratic equation before applying the formula.

Example Problem 1 (Illustrative):

Solve the quadratic equation: 2x² - 5x + 3 = 0

Steps:

1. Identify a, b, and c: In this equation, a = 2, b = -5, and c = 3.

2. Substitute into the quadratic formula:

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

3. Simplify:

   x = [5 ± √(25 - 24)] / 4 x = [5 ± √1] / 4 x = (5 ± 1) / 4

4. Find the two roots:

   x₁ = (5 + 1) / 4 = 6/4 = 3/2 x₂ = (5 - 1) / 4 = 4/4 = 1

Therefore, the solutions to the equation 2x² - 5x + 3 = 0 are x = 3/2 and x = 1.

Example Problem 2 (Involving Fractions):

Solve: (x-1)/(2x+1) + (2x+1)/(x-1) = 5/2

Steps:

1. Clear the fractions: Multiply both sides by (x-1)(2x+1) to eliminate the denominators. This will result in a quadratic equation.

2. Expand and simplify: After expanding and simplifying, you'll likely obtain an equation in the standard form ax² + bx + c = 0.

3. Identify a, b, and c: Determine the values of a, b, and c.

4. Apply the quadratic formula: Substitute the values of a, b, and c into the quadratic formula and solve for x. Remember to check your solutions to ensure they do not make the original denominators zero.

Example Problem 3 (Involving Square Roots):

Solve: √2x² + 7x + 5√2 = 0

This problem may appear intimidating due to the presence of square roots, but the approach remains the same.

1. Identify a, b, and c: Even though the coefficients contain square roots, the process of identifying a, b, and c remains the same. Here a=√2, b=7, c=5√2.

2. Apply the quadratic formula: Substitute the values into the formula. Note that you might need to simplify expressions involving square roots. Rationalizing the denominator may be necessary in some cases.

Addressing Common Challenges and Mistakes

Students often encounter difficulties while working with quadratic equations, particularly when dealing with fractions, negative coefficients, or irrational numbers. Here are some common mistakes and how to avoid them:

• Incorrect identification of a, b, and c: Double-check the values of a, b, and c before substituting them into the formula. A simple error here can lead to incorrect solutions.

• Errors in simplification: Carefully simplify the expression inside the square root and the overall fraction. Use order of operations correctly (PEMDAS/BODMAS).

• Sign errors: Pay close attention to negative signs, particularly when substituting values into the formula. A missed negative sign can drastically alter the result.

• Mistakes in calculating the discriminant: The discriminant (b² - 4ac) is crucial. Any errors in calculating it will affect the entire solution. Double-check your calculations.

• Ignoring extraneous solutions: When the equation involves fractions or square roots, always check if any of your calculated solutions make the original equation undefined (division by zero) or lead to inconsistencies. Discard these extraneous solutions.

Explanation of the Quadratic Formula's Derivation (Optional, for advanced understanding)

The quadratic formula can be derived using a method called "completing the square." This method involves manipulating the general quadratic equation to create a perfect square trinomial, which can then be factored easily. The steps are as follows:

1. Start with the general equation: ax² + bx + c = 0

2. Divide by 'a': x² + (b/a)x + c/a = 0

3. Move the constant term to the right side: x² + (b/a)x = -c/a

4. Complete the square: Add (b/(2a))² to both sides. This creates a perfect square trinomial on the left side.

5. Factor the perfect square trinomial: (x + b/(2a))² = (b² - 4ac)/(4a²)

6. Take the square root of both sides: x + b/(2a) = ±√(b² - 4ac)/(2a)

7. Solve for x: Isolate x to obtain the quadratic formula.

Frequently Asked Questions (FAQ)

• Q: What if the discriminant is negative?

  • A: A negative discriminant indicates that there are no real solutions. The roots are complex numbers, involving the imaginary unit 'i' (where i² = -1). At this level, you might focus on stating that there are no real roots.

• Q: Can I always use the quadratic formula?

  • A: Yes, the quadratic formula works for all quadratic equations, regardless of the coefficients. While factoring is sometimes quicker, the quadratic formula provides a reliable solution in all cases.

• Q: How can I check my answers?

  • A: Substitute the calculated values of 'x' back into the original quadratic equation. If the equation holds true, your solutions are correct.

• Q: What if I get a fraction as a solution?

  • A: Fractions are perfectly valid solutions. Leave your answer as a fraction in its simplest form unless the question specifies otherwise.

Conclusion

Mastering Exercise 3.4 and, more broadly, solving quadratic equations is a significant step in your mathematical journey. By understanding the quadratic formula, its application, and potential pitfalls, you'll build a strong foundation for more advanced algebraic concepts. Remember to practice regularly, paying close attention to detail and carefully checking your work. With consistent effort and the techniques outlined in this guide, you'll confidently tackle any quadratic equation you encounter. Don't hesitate to review the examples and explanations multiple times to solidify your understanding. Good luck!