Solve. (Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION.)

Expert intro: This rational equation becomes a quadratic after clearing denominators. The important final step is checking every candidate against the original equation.

Detailed walkthrough

Step 1: Note the restriction

Because the denominator is $x+2$,

$x\ne -2$

Step 2: Multiply by the LCD

Multiply both sides by $x+2$:

$\frac{2x}{x+2}(x+2)-x(x+2)=\frac{-49}{x+2}(x+2)$

So

$2x-x(x+2)=-49$

Step 3: Simplify

$2x-x^2-2x=-49$ $-x^2=-49$ $x^2=49$ $x=\pm 7$

Step 4: Check solutions

Both $x=7$ and $x=-7$ satisfy the restriction $x\ne -2$, so both work.

Answer: -7, 7

💡 Pitfall guide

Do not discard $-7$ just because the example in the prompt shows $x=7$. Always solve the equation from scratch and test both square-root answers.

🔄 Real-world variant

If the denominator were $x-2$ instead of $x+2$, the restriction would change to $x\ne 2$, but the same clearing-denominators method would still apply.

🔍 Related terms

rational equation, extraneous solution, quadratic