Math Problem Statement

Perform the following operation and express in simplest form: (x^2 - 3x - 54) / (x^2 - 9x) * (2x) / (x^2 + 8x + 12).

Solution

We are tasked with simplifying the following rational expression:

x23x54x29x2xx2+8x+12.\frac{x^2 - 3x - 54}{x^2 - 9x} \cdot \frac{2x}{x^2 + 8x + 12}.

Step-by-step Solution:

  1. Factorize each part of the expression:

    • Factorize the numerator x23x54x^2 - 3x - 54: x23x54=(x9)(x+6).x^2 - 3x - 54 = (x - 9)(x + 6).

    • Factorize the denominator x29xx^2 - 9x: x29x=x(x9).x^2 - 9x = x(x - 9).

    • Factorize the denominator x2+8x+12x^2 + 8x + 12: x2+8x+12=(x+6)(x+2).x^2 + 8x + 12 = (x + 6)(x + 2).

    Substituting these factorizations, the expression becomes: (x9)(x+6)x(x9)2x(x+6)(x+2).\frac{(x - 9)(x + 6)}{x(x - 9)} \cdot \frac{2x}{(x + 6)(x + 2)}.

  2. Simplify by canceling common factors:

    • x9x - 9 appears in both the numerator and denominator, so cancel x9x - 9.
    • x+6x + 6 appears in both the numerator and denominator, so cancel x+6x + 6.
    • xx in the numerator of the second fraction cancels with the xx in the denominator of the first fraction.

    After cancellation, we are left with: 2x+2.\frac{2}{x + 2}.

Final Answer:

2x+2.\frac{2}{x + 2}.

Note:

The expression is valid for x9,6,0,2x \neq 9, -6, 0, -2, as these values would make the original denominators equal to zero.


Do you want further clarification or details on any of these steps?

Related Questions:

  1. How do you factor trinomials like x23x54x^2 - 3x - 54?
  2. Why is it important to find restrictions on xx in rational expressions?
  3. What happens if a factor is not canceled correctly in such problems?
  4. How can you confirm your factorizations are correct?
  5. How do you approach similar problems involving division instead of multiplication?

Tip:

Always check for factors common to both the numerator and denominator before simplifying—it helps avoid mistakes!

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Math Problem Analysis

Mathematical Concepts

Algebra
Factoring
Simplification of Rational Expressions

Formulas

Factoring trinomials: ax^2 + bx + c = (px + q)(rx + s)
Simplifying rational expressions: Canceling common factors in numerators and denominators

Theorems

Zero Product Property
Properties of Rational Expressions

Suitable Grade Level

Grades 8-10