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The equation x^2 + 3x + 7 = 6/(x^2 + 3x + 2) is solved by substituting u = x^2 + 3x, resulting in a quadratic equation in u. Factoring the quadratic equation yields two possible values for u, which are then substituted back into the substitution to find the solutions for x using the quadratic formula. One case results in two real solutions, while the other yields no real solutions.

The solution to the equation \sqrt{7 + \frac{3}{\sqrt{x}}} = 7 - \frac{9}{x} is x = \frac{9}{4}.

The solution involves finding an expression for X_n, a sum of reciprocals of powers of x, using the formula for a finite geometric series. Then, the expression for x^n is used to simplify the ratio x^n / X_n, ultimately resulting in x^(n+1).

To find x, supplementary angles 5x and 4x are used, along with the fact that the sum of the interior angles of a quadrilateral is 360 degrees. Solving the equation 9x = 180 yields x = 20.

The solution demonstrates how to find the value of RS by expanding the second equation, substituting the first equation, and simplifying the resulting equation.

The solution for X in the equation 2^X + 2 = 66 is 6.

To solve for a in the equation (a/2) ÷ (5/2) = 2, first rewrite the division as multiplication by the reciprocal, then multiply the numerators and denominators, simplify the fraction, isolate a by multiplying both sides by 10, and finally solve for a by dividing both sides by 2, resulting in a = 10.

To solve the equation 16/x = x^2/4, cross-multiply to get 64 = x^3, then take the cube root of both sides to find x = 4.

The solution to the equation (1/3)^(x+3) = 9^x is x = -1.

The inverse Laplace transform of 1/x^(3/2) is 2/(sqrt(pi))*sqrt(x).