2х/х+6-144/х2-36=1

To solve the equation \( \frac{2x}{x + 6} - \frac{144}{x^2 - 36} = 1 \), we will first simplify the terms. First, notice that \( x^2 - 36 \) is a difference of squares, which can be factored: \[ x^2 - 36 = (x - 6)(x + 6) \] Now, we can rewrite the original equation: \[ \frac{2x}{x + 6} - \frac{144}{(x - 6)(x + 6)} = 1 \] Next, we find a common denominator for the left side. The common denominator is \( (x - 6)(x + 6) \). Rewriting the first term with the common denominator: \[ \frac{2x(x - 6)}{(x - 6)(x + 6)} - \frac{144}{(x - 6)(x + 6)} = 1 \] This gives us: \[ \frac{2x(x - 6) - 144}{(x - 6)(x + 6)} = 1 \] Now we can eliminate the fraction by multiplying both sides by \( (x - 6)(x + 6) \): \[ 2x(x - 6) - 144 = (x - 6)(x + 6) \] Next, simplify the right side: \[ (x - 6)(x + 6) = x^2 - 36 \] Substituting this back in gives us: \[ 2x(x - 6) - 144 = x^2 - 36 \] Expanding the left side: \[ 2x^2 - 12x - 144 = x^2 - 36 \] Now, we will move all terms to one side of the equation: \[ 2x^2 - 12x - 144 - x^2 + 36 = 0 \] This simplifies to: \[ x^2 - 12x - 108 = 0 \] Now, we can apply the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \): Where \( a = 1 \), \( b = -12 \), and \( c = -108 \). Calculating the discriminant: \[ b^2 - 4ac = (-12)^2 - 4(1)(-108) = 144 + 432 = 576 \] Now substituting back into the quadratic formula: \[ x = \frac{12 \pm \sqrt{576}}{2(1)} = \frac{12 \pm 24}{2} \] This gives us two potential solutions: 1. \( x = \frac{36}{2} = 18 \) 2. \( x = \frac{-12}{2} = -6 \) However, we need to check which solutions are valid in the context of the original equation, especially since they could create division by zero. Plugging \( x = -6 \) into the factor \( x + 6 \) shows that it's not valid (it results in division by zero). Therefore, we disregard \( x = -6 \). Thus, the only valid solution is: \[ \boxed{18} \]