Question

simplify the complex fraction.\\(dfrac{left( dfrac{8x}{x + 6}
ight)}{left( dfrac{16}{x^2 + 8x + 12}
ight)}\\)(no response) , \\(x \
eq -6, -2\\)

Explanation:

Step1: Factor the quadratic denominator

First, factor the quadratic expression \(x^2 + 8x + 12\). We need two numbers that multiply to \(12\) and add to \(8\). Those numbers are \(6\) and \(2\), so \(x^2 + 8x + 12=(x + 6)(x + 2)\).

Step2: Rewrite the complex fraction as a division

A complex fraction \(\frac{\frac{a}{b}}{\frac{c}{d}}\) is equivalent to \(\frac{a}{b}\div\frac{c}{d}\), which is the same as \(\frac{a}{b}\times\frac{d}{c}\). So our complex fraction \(\frac{\frac{8x}{x + 6}}{\frac{16}{(x + 6)(x + 2)}}\) becomes \(\frac{8x}{x + 6}\times\frac{(x + 6)(x + 2)}{16}\).

Step3: Simplify the expression

We can cancel out the common factors. The \((x + 6)\) terms cancel out, and we can simplify the coefficients and the \(x\) terms. \(8x\) and \(16\) have a common factor of \(8\), so \(\frac{8x}{16}=\frac{x}{2}\). After canceling and simplifying, we are left with \(\frac{x(x + 2)}{2}\), which expands to \(\frac{x^2 + 2x}{2}\) or we can also write it as \(\frac{x(x + 2)}{2}\). Another way: \(\frac{8x}{x + 6}\times\frac{(x + 6)(x + 2)}{16}=\frac{8x(x + 6)(x + 2)}{16(x + 6)}\). Cancel \(8\) and \(16\) (divide numerator and denominator by \(8\)) to get \(\frac{x(x + 6)(x + 2)}{2(x + 6)}\), then cancel \((x + 6)\) (since \(x
eq - 6\)) to get \(\frac{x(x + 2)}{2}=\frac{x^2+2x}{2}\) or \(\frac{x(x + 2)}{2}\).

Answer:

\(\frac{x(x + 2)}{2}\) (or \(\frac{x^2 + 2x}{2}\))