Question

2х/х+6 - 144/х^2-36=1

Answer 1

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Answer 2

To solve the equation $\frac{2x}{x+6} - \frac{144}{x^2-36} = 1$, follow these steps:

Step 1: Factor the denominators of the fractions in the equation. $x^2 - 36$ can be factored as $(x+6)(x-6)$.

Step 2: Rewrite the equation with factored denominators. $\frac{2x}{x+6} - \frac{144}{(x+6)(x-6)} = 1$.

Step 3: Find a common denominator for the fractions on the left side of the equation. The common denominator is $(x+6)(x-6)$.

Step 4: Rewrite each fraction with the common denominator. $\frac{2x(x-6)}{(x+6)(x-6)} - \frac{144}{(x+6)(x-6)} = 1$.

Step 5: Combine the fractions on the left side of the equation. $\frac{2x(x-6) - 144}{(x+6)(x-6)} = 1$.

Step 6: Distribute and simplify the numerator on the left side. $\frac{2x^2 - 12x - 144}{(x+6)(x-6)} = 1$.

Step 7: Move the denominator to the right side of the equation. $2x^2 - 12x - 144 = (x+6)(x-6)$.

Step 8: Expand the right side of the equation. $2x^2 - 12x - 144 = x^2 - 6x + 6x - 36$. $2x^2 - 12x - 144 = x^2 - 36$.

Step 9: Move all terms to one side of the equation. $2x^2 - x^2 - 12x + 36 - 144 = 0$. $x^2 - 12x - 108 = 0$.

Step 10: Factor the quadratic equation. $(x - 18)(x + 6) = 0$.

Step 11: Set each factor equal to zero and solve for $x$. $x - 18 = 0$ => $x = 18$, $x + 6 = 0$ => $x = -6$.

So, the solutions for the equation are $x = 18$ and $x = -6$.