Solve the Reciprocal Equation: Finding X in 9/x = x

$\frac{9}{x}=x$

To solve the equation $\frac{9}{x} = x$, we will eliminate the fraction by multiplying both sides by $x$, provided $x \neq 0$.

Our steps are as follows:

• Step 1: Multiply both sides by $x$ to obtain $9 = x^2$.
• Step 2: Rearrange the equation to standard quadratic form: $x^2 - 9 = 0$.
• Step 3: Recognize that this can be solved by factoring as a difference of squares: $(x - 3)(x + 3) = 0$.
• Step 4: Solve for $x$ by setting each factor to zero, giving $x - 3 = 0$ or $x + 3 = 0$.
• Step 5: Solve these equations to find $x = 3$ or $x = -3$.

Thus, the solutions to the equation $\frac{9}{x} = x$ are $x = 3$ and $x = -3$.

Therefore, the correct answer, which matches choice 3, is $x = \pm 3$.