Exercise 3.4 Maths Class 10

Exercise 3.4: A Deep Dive into Quadratic Equations (Class 10 Maths)

This article provides a thorough look to solving Exercise 3.4 in Class 10 Maths, typically focusing on the application of the quadratic formula and other methods to solve a variety of quadratic equations. Also, understanding quadratic equations is crucial for further mathematical studies, and this exercise provides valuable practice in mastering these essential concepts. We'll cover the fundamental principles, provide step-by-step solutions to representative problems, and address frequently asked questions.

Introduction to Quadratic Equations

A quadratic equation is an algebraic equation of the second degree, meaning the highest power of the variable (usually x) is 2. It is generally expressed in the standard form:

ax² + bx + c = 0

where a, b, and c are constants, and a ≠ 0 (if a were 0, it would no longer be a quadratic equation). Solving a quadratic equation means finding the values of x that satisfy the equation. There are several methods to solve these equations, each with its own advantages and disadvantages. Exercise 3.4 often focuses on applying these methods efficiently and accurately.

This part deserves a bit more attention than it usually gets.

Methods for Solving Quadratic Equations

Exercise 3.4 usually introduces or reinforces the following methods:

• Factorization: This method involves expressing the quadratic expression as a product of two linear factors. To give you an idea, x² + 5x + 6 can be factored as (x + 2)(x + 3) = 0. This leads to the solutions x = -2 and x = -3. On the flip side, factorization isn't always straightforward, especially when dealing with complex or irrational roots.

• Quadratic Formula: This is a powerful and universally applicable method. The quadratic formula provides the solutions for x directly from the coefficients a, b, and c:

  x = [-b ± √(b² - 4ac)] / 2a

  The term (b² - 4ac) is called the discriminant, often denoted as Δ (delta). The discriminant helps determine the nature of the roots:

  • Δ > 0: Two distinct real roots.
  • Δ = 0: One real root (repeated root).
  • Δ < 0: Two distinct complex roots (involving imaginary numbers).

• Completing the Square: This method involves manipulating the quadratic equation to form a perfect square trinomial, which can then be easily solved. While less frequently used directly in Exercise 3.4 compared to the quadratic formula, understanding completing the square is crucial for various mathematical concepts later on.

Step-by-Step Solutions (Illustrative Examples)

Let's tackle some sample problems representative of what you might find in Exercise 3.4. Remember that the specific questions in your exercise will vary, but the principles remain the same.

Problem 1: Solving using the Quadratic Formula

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

Solution:

Here, a = 2, b = -5, and c = 3. Applying the quadratic formula:

x = [-(-5) ± √((-5)² - 4 * 2 * 3)] / (2 * 2) x = [5 ± √(25 - 24)] / 4 x = [5 ± √1] / 4 x = (5 ± 1) / 4

This gives two solutions:

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

So, the solutions are x = 3/2 and x = 1.

Problem 2: Solving by Factorization

Solve the quadratic equation: x² - 7x + 12 = 0

Solution:

We look for two numbers that add up to -7 (the coefficient of x) and multiply to 12 (the constant term). These numbers are -3 and -4. Because of this, we can factor the equation as:

(x - 3)(x - 4) = 0

This gives the solutions:

x - 3 = 0 => x = 3 x - 4 = 0 => x = 4

So, the solutions are x = 3 and x = 4.

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Problem 3: A Problem with a Discriminant Analysis

Determine the nature of the roots of the quadratic equation: x² + 2x + 5 = 0 without solving it.

Solution:

Here, a = 1, b = 2, and c = 5. Let's calculate the discriminant:

Δ = b² - 4ac = (2)² - 4 * 1 * 5 = 4 - 20 = -16

Since Δ < 0, the quadratic equation has two distinct complex roots.

Problem 4: Word Problem Application

The area of a rectangular garden is 120 square meters. If the length is 5 meters more than the width, find the dimensions of the garden.

Solution:

Let the width be w meters. The length is (w + 5) meters. The area is given by:

w(w + 5) = 120 w² + 5w - 120 = 0

We can solve this quadratic equation using factorization or the quadratic formula. Using factorization:

(w + 15)(w - 8) = 0

This gives two possible solutions: w = -15 or w = 8. Since width cannot be negative, the width is 8 meters. And the length is w + 5 = 8 + 5 = 13 meters. Which means, the dimensions of the garden are 8 meters by 13 meters.

Explanation of Key Concepts

• Roots: The solutions of a quadratic equation are also known as its roots or zeros. These are the values of x that make the equation true.

• Discriminant: The discriminant (Δ = b² - 4ac) is a crucial indicator of the nature of the roots. Its value determines whether the roots are real and distinct, real and equal, or complex.

• Perfect Square Trinomial: A perfect square trinomial is a trinomial that can be factored into the square of a binomial. Take this: x² + 6x + 9 is a perfect square trinomial because it factors into (x + 3)². Completing the square involves manipulating a quadratic expression to create a perfect square trinomial.

Frequently Asked Questions (FAQ)

• Q: What if I can't factor the quadratic equation easily?

  • A: The quadratic formula is your best bet in such cases. It always works, regardless of whether the roots are rational, irrational, or complex.

• Q: How do I know which method to use?

  • A: If the quadratic equation is easily factorable, factorization is efficient. Otherwise, the quadratic formula is the most reliable method.

• Q: What are complex roots?

  • A: Complex roots involve the imaginary unit i, where i² = -1. They occur when the discriminant is negative.

• Q: What is the significance of the discriminant?

  • A: The discriminant tells you the nature of the roots without actually solving the equation, providing valuable information about the solutions.

• Q: Can a quadratic equation have only one root?

  • A: Yes, this happens when the discriminant is equal to zero. The root is then a repeated root.

Conclusion

Mastering the methods for solving quadratic equations is fundamental to your success in algebra and beyond. Remember to practice consistently, and don't hesitate to review the concepts if you encounter any difficulties. By understanding the principles and practicing with various examples, you'll develop the skills and confidence needed to tackle more complex mathematical problems in the future. Now, exercise 3. 4 provides valuable practice in applying these methods, from factorization and the quadratic formula to understanding the significance of the discriminant. With dedicated effort, you'll master quadratic equations and build a strong foundation for your mathematical journey.