Class 9 Maths Exercise 4.7: Factorization & Algebra

Class 9 Mathematics – Exercise 4.7: Factorization and Algebraic Manipulation (Square Root Methods)

In this lesson, students learn how to find square roots of algebraic expressions using advanced factorization techniques. Exercise 4.7 focuses on recognizing perfect square patterns, simplifying polynomials, and applying systematic methods such as factorization, substitution, and the division method.

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Understanding Square Roots of Algebraic Expressions

The main objective of this exercise is to convert algebraic expressions into perfect square forms. A perfect square trinomial follows the identity:

(a + b)² = a² + 2ab + b²

(a - b)² = a² - 2ab + b²

If an expression matches these patterns, its square root becomes the binomial inside the square.

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Method 1: Factorization Approach

In factorization, we rewrite expressions as repeated binomials.

Example 1

Expression: 16y² - 56y + 49

• 16y² = (4y)²
• 49 = 7²
• Middle term = 2 × 4y × 7 = 56y

Since the middle term is negative, the expression becomes:

(4y - 7)²

Final answer: 4y - 7

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Example 2

Expression: 25a⁴ - 30a³ + 9a²

First, take common factor:

a²(25a² - 30a + 9)

Now factor inside bracket:

(5a - 3)²

Final answer: a(5a - 3)

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Method 2: Substitution Technique

Substitution simplifies complex expressions by replacing repeated terms with a single variable.

Example

Expression: (x² - 1/x²)² + 4(x² - 1/x²) + 4

Let:

a = x² - 1/x²

Now expression becomes:

a² + 4a + 4

Factorizing:

(a + 2)²

Final answer:

x² - 1/x² + 2

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Method 3: Division Method

This method is used for higher-degree polynomials where factorization is not obvious.

Example 1

Expression: x⁴ + 8x³ + 20x² + 16x + 4

Step-by-step division gives:

x² + 4x + 2

This shows that the original expression is a perfect square of (x² + 4x + 2).

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Example 2

Expression: x⁴ + 10x³ + 31x² + 30x + 9

Using the same method, the final result becomes:

x² + 5x + 3

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Why This Topic Is Important

Exercise 4.7 strengthens algebraic reasoning by training students to identify hidden patterns in polynomials. It is a foundation for:

• Quadratic equations
• Advanced factorization
• Problem-solving in exams

Students who master this topic can solve algebra problems faster and more accurately in board exams.

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Key Exam Tips

1. Always check for common factors first.
2. Identify perfect square patterns quickly.
3. Use substitution for repeated expressions.
4. Verify answers by expanding back.
5. Practice different methods for the same question.

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Mastering this exercise improves both speed and accuracy in algebra. It builds the logical foundation required for higher-level mathematics.