Question

simplify the expression completely if possible. \\(\frac{2x^2 + 16x}{x^2 - 49}\\)

Explanation:

Step1: Factor numerator and denominator

Factor the numerator \(2x^2 + 16x\) by taking out the greatest common factor, which is \(2x\). So, \(2x^2 + 16x = 2x(x + 8)\).

Factor the denominator \(x^2 - 49\) using the difference of squares formula \(a^2 - b^2 = (a + b)(a - b)\). Here, \(a = x\) and \(b = 7\), so \(x^2 - 49 = (x + 7)(x - 7)\).

Now the expression becomes \(\frac{2x(x + 8)}{(x + 7)(x - 7)}\).

Step2: Check for common factors

There are no common factors between the numerator \(2x(x + 8)\) and the denominator \((x + 7)(x - 7)\) that can be canceled out. So the expression is already in its simplest form.

Answer:

\(\frac{2x(x + 8)}{(x + 7)(x - 7)}\) (or expanded numerator as \(\frac{2x^2 + 16x}{x^2 - 49}\) is also acceptable if no further simplification is possible, but factored form is more simplified)