2+x=3/x

Amanita, let's solve the equation $2 + x = \frac{3}{x}$ step by step. Here’s our plan:

Step 1: Move all terms to one side to set the equation to zero.
Step 2: Multiply both sides by $x$ to eliminate the fraction.
Step 3: Rearrange the equation into standard quadratic form.
Step 4: Solve the quadratic equation for $x$.
Step 5: Check the solutions to make sure they are valid (since $x$ cannot be zero).

Let's begin!

Step 1:
Start with the equation:

Subtract $\frac{3}{x}$ from both sides:

Step 2:
Multiply both sides by $x$ to get rid of the fraction:

$x(2 + x - \frac{3}{x}) = 0$

This simplifies to:

Step 3:
Rearrange the equation into standard quadratic form:

Step 4:
Now, let's solve the quadratic equation $x^2 + 2x - 3 = 0$.

Amanita, after solving the quadratic equation, we find two possible solutions: $x = -3$ and $x = 1$.

Step 5:
Let's check if both solutions are valid by substituting them back into the original equation $2 + x = \frac{3}{x}$:

• For $x = -3$: $2 + (-3) = -1$ and $\frac{3}{-3} = -1$, so this works.
• For $x = 1$: $2 + 1 = 3$ and $\frac{3}{1} = 3$, so this also works.

So, the solutions to the equation $2 + x = \frac{3}{x}$ are:

$\boxed{x = -3 \text{ and } x = 1}$