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

Key takeaway: This is a rational equation with one denominator, so the fastest method is to multiply every term by the LCD and reduce it to a polynomial equation.

Step 1: Find the restriction

The denominator is $x+2$, so

$x\ne -2$

Step 2: Multiply by the LCD

Multiply both sides by $x+2$:

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

This simplifies to

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

Step 3: Expand and solve

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

Step 4: Check the restriction

Neither $7$ nor $-7$ makes $x+2=0$, so both are valid.

Answer: -7, 7

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Pitfalls the pros know 👇 Do not forget the restriction $x\ne -2$. Also, when you solve $x^2=49$, remember that square roots give two answers: $x=7$ and $x=-7$.

What if the problem changes? If the right-hand side were different, the same LCD method would still work. The final step would depend on whether the reduced equation factors, becomes quadratic, or has no real solutions.

Tags: LCD, quadratic equation, restriction