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((x+3)/(4x^2-9))/((x^2+7x+12)/(2x^2+7x-15))

3 min read Jun 16, 2024
((x+3)/(4x^2-9))/((x^2+7x+12)/(2x^2+7x-15))

Simplifying Complex Fractions: A Step-by-Step Guide

This article will guide you through simplifying the complex fraction:

((x+3)/(4x^2-9))/((x^2+7x+12)/(2x^2+7x-15))

1. Factor all the polynomials:

  • Numerator of the main fraction: (x+3) is already factored.
    • 4x^2-9 is a difference of squares, factored as (2x+3)(2x-3)
  • Denominator of the main fraction:
    • x^2+7x+12 factors as (x+3)(x+4)
    • 2x^2+7x-15 factors as (2x-3)(x+5)

2. Rewrite the complex fraction as division:

The fraction bar in the middle of the expression signifies division. We can rewrite the expression as:

((x+3)/(2x+3)(2x-3)) ÷ ((x+3)(x+4)/(2x-3)(x+5))

3. Invert the second fraction and multiply:

Dividing by a fraction is the same as multiplying by its inverse. This means we flip the second fraction and multiply:

((x+3)/(2x+3)(2x-3)) × ((2x-3)(x+5)/(x+3)(x+4))

4. Cancel out common factors:

Notice that (x+3) and (2x-3) appear in both the numerator and denominator. These factors cancel out:

1/(2x+3) × (x+5)/(x+4)

5. Simplify the final expression:

Multiplying the remaining fractions gives us the final simplified form:

(x+5)/(2x+3)(x+4)

Therefore, the simplified form of the complex fraction ((x+3)/(4x^2-9))/((x^2+7x+12)/(2x^2+7x-15)) is (x+5)/(2x+3)(x+4)

Important Note: This solution assumes that the values of x that make the denominator zero are excluded. This is because division by zero is undefined.

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