Math Problem Statement

Simplify the expression completely if possible. start fraction, 2, x, squared, plus, 16, x, divided by, x, squared, minus, 49, end fraction x 2 −49 2x 2 +16x ​

Solution

We are tasked with simplifying the expression:

2x2+16xx249\frac{2x^2 + 16x}{x^2 - 49}

Step 1: Factor the numerator and the denominator.

  • Numerator: 2x2+16x2x^2 + 16x

    • Factor out the greatest common factor (GCF) of 2x:

    2x2+16x=2x(x+8)2x^2 + 16x = 2x(x + 8)

  • Denominator: x249x^2 - 49

    • This is a difference of squares, and it can be factored as:

    x249=(x+7)(x7)x^2 - 49 = (x + 7)(x - 7)

Step 2: Substitute the factored forms into the original expression:

2x(x+8)(x+7)(x7)\frac{2x(x + 8)}{(x + 7)(x - 7)}

Step 3: Simplify the expression.

  • There are no common factors between the numerator and the denominator, so the expression is already in its simplest form.

Thus, the simplified expression is:

2x(x+8)(x+7)(x7)\frac{2x(x + 8)}{(x + 7)(x - 7)}


Would you like to see a detailed breakdown of any part of this solution? Or is there another math topic you'd like to explore?

Here are 5 related questions for further study:

  1. How would you factor a trinomial like x2+6x+8x^2 + 6x + 8?
  2. What are other ways to simplify rational expressions?
  3. How do you determine when to cancel common factors in rational expressions?
  4. What are the conditions for division by zero in rational expressions?
  5. How would you approach simplifying more complex rational expressions involving polynomials of higher degrees?

Tip: Always look for the greatest common factor (GCF) first when factoring expressions, as it can simplify the process significantly.

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Math Problem Analysis

Mathematical Concepts

Algebra
Factoring
Rational Expressions
Difference of Squares

Formulas

Factoring of a trinomial
Difference of squares formula: a^2 - b^2 = (a + b)(a - b)

Theorems

Difference of squares theorem
Factoring trinomials

Suitable Grade Level

Grades 8-10