Math Problem Statement

8x/x+4 + x+4/12x^2

Solution

The expression you've written is:

8xx+4+x+412x2\frac{8x}{x+4} + \frac{x+4}{12x^2}

To simplify this, we can follow these steps:

Step 1: Find a common denominator

The denominators in the expression are x+4x+4 and 12x212x^2. The least common denominator (LCD) between these is 12x2(x+4)12x^2(x+4).

Step 2: Rewrite each fraction with the LCD

Now, we rewrite each term so that both have the same denominator:

  1. 8xx+4\frac{8x}{x+4} becomes: 8x12x2(x+4)12x2=96x312x2(x+4)\frac{8x \cdot 12x^2}{(x+4) \cdot 12x^2} = \frac{96x^3}{12x^2(x+4)}

  2. x+412x2\frac{x+4}{12x^2} remains as: x+412x2(x+4)\frac{x+4}{12x^2(x+4)}

Step 3: Combine the two fractions

Now that they have the same denominator, we can combine them: 96x3+(x+4)12x2(x+4)\frac{96x^3 + (x+4)}{12x^2(x+4)}

Simplifying the numerator: 96x3+x+496x^3 + x + 4

Thus, the expression becomes: 96x3+x+412x2(x+4)\frac{96x^3 + x + 4}{12x^2(x+4)}

This is the simplified form of the original expression. If you need to factor further or analyze the behavior of this expression, you could do so depending on the context.

Would you like a detailed explanation of any specific part?

Follow-up questions:

  1. What are the critical values of xx where the expression is undefined?
  2. How does the expression behave as xx approaches zero?
  3. Can this expression be simplified further by factoring?
  4. What is the value of this expression for a specific value of xx, like x=2x = 2?
  5. Can this expression be applied in a real-world context, such as in physics?

Tip:

Always check for undefined points by examining the denominators for values of xx that would make them zero.

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Math Problem Analysis

Mathematical Concepts

Algebra
Rational Expressions
Common Denominators
Simplification

Formulas

Least Common Denominator (LCD)
Fraction Addition Formula

Theorems

Rational Expression Simplification Theorem
Undefined Points in Rational Functions

Suitable Grade Level

Grades 9-11