Question

solve for w.
\\(\frac{18}{w} = w + \frac{2}{w}\\)
if there is more than one solution, separate them with commas.
if there is no solution, click on
o solution\.

Explanation:

Step1: Eliminate the denominator

Multiply both sides of the equation \(\frac{18}{w}=w + \frac{2}{w}\) by \(w\) (note that \(w
eq0\) to avoid division by zero) to get \(18 = w^{2}+2\).

Step2: Rearrange the equation

Subtract 18 from both sides of the equation \(18 = w^{2}+2\) to obtain a quadratic equation: \(w^{2}+2 - 18=0\), which simplifies to \(w^{2}-16 = 0\).

Step3: Solve the quadratic equation

We can factor the quadratic equation \(w^{2}-16 = 0\) using the difference of squares formula \(a^{2}-b^{2}=(a + b)(a - b)\), where \(a = w\) and \(b = 4\). So, \((w + 4)(w - 4)=0\).
Setting each factor equal to zero gives us two equations: \(w+4 = 0\) or \(w - 4=0\).
Solving \(w+4 = 0\) gives \(w=-4\), and solving \(w - 4=0\) gives \(w = 4\). We need to check these solutions in the original equation.
For \(w = 4\): Left - hand side of the original equation \(\frac{18}{4}=\frac{9}{2}\), right - hand side \(4+\frac{2}{4}=4+\frac{1}{2}=\frac{9}{2}\). So, \(w = 4\) is a valid solution.
For \(w=-4\): Left - hand side of the original equation \(\frac{18}{-4}=-\frac{9}{2}\), right - hand side \(-4+\frac{2}{-4}=-4-\frac{1}{2}=-\frac{9}{2}\). So, \(w=-4\) is also a valid solution.

Answer:

• 4, 4