Class 10 Maths Exercise 4.4

Conquer Class 10 Maths Exercise 4.4: Quadratic Equations

Exercise 4.4 in most Class 10 mathematics textbooks focuses on solving quadratic equations using the quadratic formula. This exercise builds upon previous lessons where you've learned about different methods to solve quadratic equations, including factorization and completing the square. Understanding the quadratic formula is crucial, as it provides a universal method for solving any quadratic equation, regardless of whether it's easily factorable. This article will guide you through Exercise 4.4, providing detailed explanations, examples, and tips to help you master this important topic.

Understanding Quadratic Equations

Before we dive into the solutions, let's refresh our understanding of quadratic equations. A quadratic equation is an equation of 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). 'x' is the variable we need to solve for. The solutions, also known as roots of the equation, represent the values of 'x' that make the equation true.

The Quadratic Formula: Your Key to Success

The quadratic formula provides a direct method to find the roots of a quadratic equation. It's derived from the method of completing the square and is a powerful tool in your mathematical arsenal. The formula is:

x = [-b ± √(b² - 4ac)] / 2a

Let's break this down:

• -b: The negative of the coefficient of 'x'.
• ±: This symbol indicates that there are two possible solutions: one using the positive square root and the other using the negative square root.
• √(b² - 4ac): This part is called the discriminant. It determines the nature of the roots. We'll discuss this in more detail later.
• 2a: Twice the coefficient of x².

Step-by-Step Guide to Solving Quadratic Equations using the Quadratic Formula

Here's a step-by-step guide to solving quadratic equations using the quadratic formula, mirroring the approach likely used in Exercise 4.4:

1. Identify a, b, and c: Write the quadratic equation in the standard form ax² + bx + c = 0. Then, identify the values of 'a', 'b', and 'c'. For example, in the equation 2x² + 5x - 3 = 0, a = 2, b = 5, and c = -3.

2. Substitute into the Quadratic Formula: Substitute the values of 'a', 'b', and 'c' into the quadratic formula: x = [-b ± √(b² - 4ac)] / 2a.

3. Calculate the Discriminant (b² - 4ac): First, calculate the discriminant. This will tell you about the nature of the roots.

   • If b² - 4ac > 0, there are two distinct real roots.
   • If b² - 4ac = 0, there is one real root (a repeated root).
   • If b² - 4ac < 0, there are no real roots (the roots are complex numbers, involving the imaginary unit 'i'). At the Class 10 level, you'll likely only encounter real roots.

4. Solve for x: Substitute the value of the discriminant back into the quadratic formula and simplify the expression. Remember to consider both the positive and negative square roots to find both solutions for 'x'.

5. Check your Solutions: Substitute each value of 'x' back into the original quadratic equation to verify that they are indeed the correct roots.

Worked Examples

Let's work through a few examples similar to those found in Exercise 4.4:

Example 1: Solve the equation 2x² - 5x + 3 = 0

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

2. Substitute into the Quadratic Formula: x = [5 ± √((-5)² - 4 * 2 * 3)] / (2 * 2)

3. Calculate the Discriminant: b² - 4ac = 25 - 24 = 1

4. Solve for x: x = [5 ± √1] / 4 This gives two solutions:

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

5. Check: Substitute x = 3/2 and x = 1 back into the original equation to verify that they are correct.

Example 2: Solve the equation x² - 4x + 4 = 0

1. Identify a, b, and c: a = 1, b = -4, c = 4

2. Substitute into the Quadratic Formula: x = [4 ± √((-4)² - 4 * 1 * 4)] / (2 * 1)

3. Calculate the Discriminant: b² - 4ac = 16 - 16 = 0

4. Solve for x: x = [4 ± √0] / 2 = 4/2 = 2 There is only one real root (a repeated root).

5. Check: Substitute x = 2 back into the original equation.

Example 3: Dealing with Irrational Roots

Solve the equation x² - 2x - 2 = 0

1. Identify a, b, and c: a = 1, b = -2, c = -2

2. Substitute into the Quadratic Formula: x = [2 ± √((-2)² - 4 * 1 * -2)] / (2 * 1)

3. Calculate the Discriminant: b² - 4ac = 4 + 8 = 12

4. Solve for x: x = [2 ± √12] / 2 = [2 ± 2√3] / 2 = 1 ± √3

This results in two irrational roots: x = 1 + √3 and x = 1 - √3. Leaving the answer in this surd form is perfectly acceptable.

The Discriminant: A Deeper Dive

The discriminant (b² - 4ac) provides valuable information about the nature of the roots of the quadratic equation. As mentioned earlier:

• b² - 4ac > 0: Two distinct real roots. The parabola intersects the x-axis at two different points.
• b² - 4ac = 0: One real root (repeated root). The parabola touches the x-axis at one point. This represents a perfect square trinomial.
• b² - 4ac < 0: No real roots. The parabola does not intersect the x-axis. The roots are complex numbers (involving 'i', the imaginary unit).

Frequently Asked Questions (FAQs)

• Q: What if I can factor the quadratic equation easily? Should I still use the quadratic formula? A: While the quadratic formula works for all quadratic equations, factorization is often quicker and easier if the equation is easily factorable. Choose the method that you find most efficient and comfortable.

• Q: What if I get a negative number under the square root? A: At the Class 10 level, you'll typically only encounter quadratic equations with real roots. A negative discriminant indicates that there are no real solutions; the roots are complex numbers which are covered in higher-level mathematics.

• Q: How can I improve my accuracy in solving these problems? A: Practice is key! Work through many problems from your textbook and other sources. Double-check your calculations at each step to avoid errors. Understanding the order of operations (PEMDAS/BODMAS) is crucial for correct calculations.

Conclusion

Mastering the quadratic formula is a significant achievement in your Class 10 maths journey. It's a fundamental tool that will serve you well in future mathematical studies. By understanding the steps involved, practicing regularly, and utilizing the insights provided by the discriminant, you'll confidently tackle Exercise 4.4 and beyond. Remember to break down complex problems into smaller, manageable steps, and don't hesitate to review the fundamental concepts if you encounter any difficulties. With consistent effort and a methodical approach, you'll find success in conquering quadratic equations. Good luck!