Class 9 Maths Exercise 5.2: Linear Equations & Inequalities

Class 9 Mathematics Exercise 5.2 – Radical Equations Explained Step-by-Step

Exercise 5.2 from Class 9 Mathematics introduces students to one of the most important algebraic concepts: radical equations. These equations contain variables inside square roots and require careful algebraic manipulation to solve correctly.

Many students find radicals difficult at first because they involve roots, squaring, and multiple algebraic steps. However, once you understand the correct method, these questions become straightforward and logical.

In this lesson, you will learn how to solve radical equations step-by-step, how to isolate square roots properly, and how to verify your answers accurately for FBISE, Matric, and O-Level examinations.

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What Are Radical Equations?

A radical equation is an equation in which the variable appears inside a radical sign such as a square root.

√(x + 5) = 7

In the equation above, the variable x is inside the square root, so this is a radical equation.

The main technique used to solve these equations is:

• Isolate the radical
• Square both sides
• Simplify the equation
• Solve for the variable
• Verify the answer

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Why Squaring Both Sides Works

The square root and square operations cancel each other:

(√x)² = x

That is why we square both sides of the equation to remove the radical sign.

However, students often make mistakes by squaring too early without isolating the radical first. Always move constants before squaring.

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Example 1 – Basic Radical Equation

√(2x) = 4

Step 1: Square both sides

(√2x)² = 4²

2x = 16

Step 2: Divide by 2

x = 8

This is the final answer.

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Example 2 – Radical Equation with a Constant

√(y + 4) − 3 = 2

First isolate the radical by moving −3 to the right side.

√(y + 4) = 5

Now square both sides:

y + 4 = 25

y = 21

Important Tip: Never square the equation before isolating the square root. This is one of the most common mistakes in examinations.

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Example 3 – Radicals on Both Sides

√(5t − 2) = √(3t + 4)

Since both sides contain square roots, square both sides directly.

5t − 2 = 3t + 4

Rearrange the equation:

5t − 3t = 4 + 2

2t = 6

t = 3

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Example 4 – Fractional Radical Equation

√((a + 6)/(a + 2)) = √((a + 2)/(a − 1))

First square both sides to remove radicals:

(a + 6)/(a + 2) = (a + 2)/(a − 1)

Cross multiply:

(a + 6)(a − 1) = (a + 2)(a + 2)

Expand both sides:

a² + 5a − 6 = a² + 4a + 4

Simplify:

5a − 4a = 4 + 6

a = 10

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Checking the Solution

Always verify your answer by putting the value back into the original equation.

This step is extremely important because squaring both sides can sometimes create invalid solutions called extraneous roots.

Students who skip verification often lose marks even when most steps are correct.

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Common Mistakes Students Make

• Squaring before isolating the radical
• Forgetting to square both sides
• Making sign errors during simplification
• Skipping verification
• Incorrect expansion during cross multiplication

Careful step-by-step working helps avoid these mistakes.

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Why Exercise 5.2 Is Important

Radical equations build the foundation for advanced algebra and higher mathematics. These concepts are later used in:

• Coordinate Geometry
• Quadratic Equations
• Functions and Graphs
• Physics formulas involving roots
• Engineering calculations

Strong understanding of radicals also improves logical thinking and algebraic manipulation skills.

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Exam Preparation Tips

• Write every algebraic step clearly
• Practice squaring carefully
• Revise identities and factorization
• Check all final answers
• Practice textbook and past paper questions regularly

Students who practice radical equations consistently usually find higher algebra chapters much easier later in Class 9 and Class 10.

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Students Also Search For

• Class 9 Maths Exercise 5.2 Solutions
• Radical Equations Class 9 Notes
• Square Root Equations Solved Examples
• FBISE Class 9 Algebra Solutions
• Past Paper Questions on Radical Equations
• STEMBridge Mathematics Lectures

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Related Posts

• Class 9 Mathematics - Measure of Central Tendency
• Class 9 Mathematics - Exercise 11.2 Basic Statistics
• Class 9 Mathematics - Statistics (Theoretical Part)

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