Question

Consider this rational equation.二+-==Use the least common denominator to simplify the rational equation into a standard form quadratic equation.Replace the values of b and to create the equation.

Answer 1

ANSWER

$\begin{equation*} x^2-10x+8=0 \end{equation*}$

EXPLANATION

We want to rewrite the rational equation as a quadratic equation:

$\frac{1}{x}+\frac{1}{x-2}=\frac{1}{4}$

First, let us find the lowest common denominator for the left-hand side of the equation:

$\begin{gathered} \frac{1}{x}+\frac{1}{x-2}=\frac{1}{4} \\ \\ \frac{x-2+x}{x(x-2)}=\frac{1}{4} \\ \\ \frac{2x-2}{x^2-2x}=\frac{1}{4} \end{gathered}$

Next, cross-multiply:

$\begin{gathered} 4(2x-2)=x^2-2x \\ \\ 8x-8=x^2-2x \end{gathered}$

Finally, move all the terms to the left-hand side and simplify:

$\begin{gathered} 8x-8-x^2+2x=0 \\ \\ -x^2+2x+8x-8=0 \\ \\ -x^2+10x-8=0 \\ \\ \Rightarrow x^2-10x+8=0 \end{gathered}$

That is the answer.