Math Problem Statement

Simplify the rational expression: (x^2 - 4) / (x^2 + x - 6) ÷ (2x^2 - x - 10) / (x^2 + 8x + 15) × (2x^2 + x - 15) / (x^2 - 25)

Solution

This is an equation involving rational expressions. We can simplify it step by step.

We are given:

x24x2+x6=2x2x10x2+8x+152x2+x15x225\frac{x^2 - 4}{x^2 + x - 6} = \frac{2x^2 - x - 10}{x^2 + 8x + 15} \cdot \frac{2x^2 + x - 15}{x^2 - 25}

Step 1: Factor the quadratic expressions

  • The numerator x24x^2 - 4 is a difference of squares, so it factors as: x24=(x2)(x+2)x^2 - 4 = (x - 2)(x + 2)

  • The denominator x2+x6x^2 + x - 6 factors as: x2+x6=(x2)(x+3)x^2 + x - 6 = (x - 2)(x + 3)

  • The numerator 2x2x102x^2 - x - 10 factors as: 2x2x10=(2x+5)(x2)2x^2 - x - 10 = (2x + 5)(x - 2)

  • The denominator x2+8x+15x^2 + 8x + 15 factors as: x2+8x+15=(x+3)(x+5)x^2 + 8x + 15 = (x + 3)(x + 5)

  • The numerator 2x2+x152x^2 + x - 15 factors as: 2x2+x15=(2x3)(x+5)2x^2 + x - 15 = (2x - 3)(x + 5)

  • The denominator x225x^2 - 25 is a difference of squares, so it factors as: x225=(x5)(x+5)x^2 - 25 = (x - 5)(x + 5)

Step 2: Substitute the factored terms into the equation

Now, substitute the factored expressions into the equation:

(x2)(x+2)(x2)(x+3)=(2x+5)(x2)(x+3)(x+5)(2x3)(x+5)(x5)(x+5)\frac{(x - 2)(x + 2)}{(x - 2)(x + 3)} = \frac{(2x + 5)(x - 2)}{(x + 3)(x + 5)} \cdot \frac{(2x - 3)(x + 5)}{(x - 5)(x + 5)}

Step 3: Cancel common factors

We can cancel the common factors in the numerator and denominator across both sides:

  • On the left-hand side, (x2)(x - 2) cancels from both the numerator and denominator.
  • On the right-hand side, (x+5)(x + 5) cancels in two places, and (x2)(x - 2) cancels from the numerator and denominator.

After canceling, we are left with:

x+2x+3=(2x+5)(2x3)(x5)(x+3)\frac{x + 2}{x + 3} = \frac{(2x + 5)(2x - 3)}{(x - 5)(x + 3)}

Step 4: Cross multiply to simplify further

Now, cross-multiply to solve for xx:

(x+2)(x5)(x+3)=(2x+5)(2x3)(x+3)(x + 2)(x - 5)(x + 3) = (2x + 5)(2x - 3)(x + 3)

From here, we can expand both sides and solve for xx. Would you like me to continue solving this equation or would you like further steps?

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Math Problem Analysis

Mathematical Concepts

Algebra
Rational Expressions
Factoring Quadratic Equations
Simplifying Expressions

Formulas

Difference of squares: a^2 - b^2 = (a - b)(a + b)
Factoring quadratics: ax^2 + bx + c = (mx + n)(px + q)

Theorems

Cross Multiplication for Equations
Properties of Rational Expressions

Suitable Grade Level

Grades 9-11