To solve this equation, we first need to find a common denominator for the fractions on the left side of the equation. The common denominator for (x-2) and 4 is 4(x-2), so we rewrite the equation as:
4(x+7)/(4(x-2)) + 5(x-2)/(4(x-2)) = 36/(x^2-4)
Now, we combine the fractions on the left side of the equation:
(4(x+7) + 5(x-2))/(4(x-2)) = 36/(x^2-4)
Expand the numerator on the left side:
(4x + 28 + 5x - 10)/(4(x-2)) = 36/(x^2-4)
Combine like terms:
(9x + 18)/(4x - 8) = 36/(x^2-4)
Now, cross multiply to get rid of the fractions:
(9x + 18)(x^2 - 4) = 36(4x - 8)
Expand both sides:
9x^3 - 36x + 18x^2 - 72 = 144x - 288
Combine like terms on the left side:
9x^3 + 18x^2 - 36x - 72 = 144x - 288
Subtract 144x and add 288 to both sides:
9x^3 + 18x^2 - 180x - 216 = 0
Now, we have a cubic equation that can be solved by factoring, using the rational root theorem, or by numerical methods.
To solve this equation, we first need to find a common denominator for the fractions on the left side of the equation. The common denominator for (x-2) and 4 is 4(x-2), so we rewrite the equation as:
4(x+7)/(4(x-2)) + 5(x-2)/(4(x-2)) = 36/(x^2-4)
Now, we combine the fractions on the left side of the equation:
(4(x+7) + 5(x-2))/(4(x-2)) = 36/(x^2-4)
Expand the numerator on the left side:
(4x + 28 + 5x - 10)/(4(x-2)) = 36/(x^2-4)
Combine like terms:
(9x + 18)/(4x - 8) = 36/(x^2-4)
Now, cross multiply to get rid of the fractions:
(9x + 18)(x^2 - 4) = 36(4x - 8)
Expand both sides:
9x^3 - 36x + 18x^2 - 72 = 144x - 288
Combine like terms on the left side:
9x^3 + 18x^2 - 36x - 72 = 144x - 288
Subtract 144x and add 288 to both sides:
9x^3 + 18x^2 - 180x - 216 = 0
Now, we have a cubic equation that can be solved by factoring, using the rational root theorem, or by numerical methods.