Finding HCF & LCM by Factorization Method

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Algebraic Manipulations

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1. Simplifying Expressions:

Simplifying expressions involves combining like terms, removing parentheses, and using the distributive property.

Example 1:

Simplify the expression 2x + 3y + x – 2y.

Solution:

Use the distributive property: 2(a + 3) = 2a + 6, and -4(2a – 1) = -8a + 4.

Combine like terms: 2a – 8a = -6a.

The simplified expression is -6a + 6.

Example 2:

Simplify the expression 2(a + 3) – 4(2a – 1).

Solution:

Use the distributive property: 2(a + 3) = 2a + 6, and -4(2a – 1) = -8a + 4.

Combine like terms: 2a – 8a = -6a.

The simplified expression is -6a + 6.

2. Solving Equations:

Solving equations involves finding the value of the variable that makes the equation true.

Examples 1:

Solve the equation 2x – 5 = 9.

Solution:

Add 5 to both sides to isolate the term with x: 2x = 14.

Divide by 2 to solve for x: x = 7.

Examples 2:

Solve the equation 3(x – 2) = 12.

Solution:

Use the distributive property: 3(x – 2) = 3x – 6.

Add 6 to both sides to isolate the term with x: 3x = 18.

Divide by 3 to solve for x: x = 6.

3. Factoring Expressions:

Factoring involves expressing an expression as a product of its factors.

Example1:

Factor the expression x² – 4.

Solution:

Recognize it as a difference of squares: x² – 4 = (x + 2) (x – 2).

Example 2:

Factor the expression 2x² + 8x.

Solution:

Factor out the common factor: 2x² + 8x = 2x (x + 4).

4. Simplifying Fractions:

Simplifying fractions involves canceling out common factors in the numerator and denominator.

Example1:

Simplify the fraction (4x² – 16) / (2x² – 8).

Solution:

Factor the numerator and denominator: (4x² – 16) / (2x² – 8) = 4(x + 4) / 2(x – 4).

Cancel out common factors: 4(x + 4) / 2(x – 4) = 2(x + 4) / (x – 4).

Example 2:

Simplify the fraction (6x³ – 3x) / (9x² – 6).

Solution:

Factor the numerator and denominator: (6x³ – 3x) / (9x² – 6) = 3x (2x² – 1) / 3(3x² – 2).

Cancel out common factors: 3x (2x² – 1) / 3(3x² – 2) = x (2x² – 1) / (3x² – 2).