Simplify the complex fraction.
-$$\frac{\frac{4}{x}-\frac{x}{4}}{\frac{1}{4}-\frac{1}{x}}$$
A) $-(x-4)$
B) $4 x(x+4)$
C) $x+4$
D) $-(x+4)$

Question

Quizplus · Accepted Answer

To simplify the given complex fraction, we first simplify the numerator and the denominator separately. Numerator: $\frac{4}{x} - \frac{x}{4} = \frac{16 - x^2}{4x}$Denominator: $\frac{1}{4} - \frac{1}{x} = \frac{x - 4}{4x}$Now, the complex fraction becomes: $\frac{\frac{16 - x^2}{4x}}{\frac{x - 4}{4x}}$Simplifying further by multiplying the numerator by the reciprocal of the denominator gives: $\frac{16 - x^2}{4x} \cdot \frac{4x}{x - 4}$The $4x$ terms cancel out, leaving: $\frac{16 - x^2}{x - 4}$Factoring the numerator as a difference of squares: $\frac{-(x^2 - 16)}{x - 4} = \frac{-(x + 4)(x - 4)}{x - 4}$Canceling the $(x - 4)$ terms gives: $-(x + 4)$Therefore, the simplified form of the given complex fraction is $-(x + 4)$, which corresponds to choice D.

Simplify the Complex Fraction 4x−x414−1x\frac{\frac{4}{x}-\frac{x}{4}}{\frac{1}{4}-\frac{1}{x}}41​−x1​x4​−4x​​ A) −(x−4)-(x-4)−(x−4) B) 4x(x+4)4 x(x+4)4x(x+4) C) x+4x+4x+4 D) −(x+4)-(x+4)−(x+4)

Simplify the Complex Fraction $\frac{\frac{4}{x}-\frac{x}{4}}{\frac{1}{4}-\frac{1}{x}}$ A) $-(x-4)$ B) $4 x(x+4)$ C) $x+4$ D) $-(x+4)$