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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).

The Laplace transform of sin^2(t) is 2 / (s * (s^2 + 4)).

The Laplace transform of e^(t/2) is 1/(s - 1/2), provided s > 1/2.

The equation |a - 5| = |5 - a| is proven by considering two cases: when (a - 5) is greater than or equal to zero and when (a - 5) is less than zero. In both cases, the equation holds true, demonstrating that the absolute value of the difference between a and 5 is equal to the absolute value of the difference between 5 and a for all values of a.

The rational function 1/(3x^2 + 4x + 1) is decomposed into partial fractions as (3/2)/(3x + 1) - (1/2)/(x + 1) by factoring the denominator and solving for the unknown constants using strategic values of x.

The partial fraction decomposition of the rational expression x/((x+1)(x-4)) is (1/5)/(x+1) + (4/5)/(x-4).

The area of the region defined by the inequalities y ≥ x, y > -x + 1, and y < 10 is 90.25 square units.

The system of equations is solved by substitution, expanding, simplifying, using the quadratic formula to find x values, and then substituting back into the equations to find the corresponding y values. The solutions are (3, 4) and (-4, -3).

The solution to the system of equations x + 2y = 5 and 3x - y = 1 is x = 1 and y = 2, found using the elimination method.

The solution for x in the equation involving fractions with (x-1) and (x+1) in the denominator is x = √2 and x = -√2.

The solution to the equation 3^(a^2 - 10a + 25)! = 729^4 is a = 3 or a = 7, found by equating exponents after expressing both sides with the same base and solving the resulting quadratic equation.

To find the difference between ba and ab, where a and b are digits, we analyze the units and hundreds digits, leading to the conclusion that the hundreds digit of the difference is 3.

The solution to the equation 4a³ = 2⁵ is a = 2.

The equation was solved by simplifying the square root, isolating the square root term, squaring both sides, moving all terms to one side, factoring out a common factor, and solving for x, resulting in two solutions: x = 0 and x = 16/81.

The solution to the equation 3^x = 1 and 3^y = 81 is x + y = 4.

By solving the system of equations x + y = 3 and x - y = 9, we find x = 6 and y = -3. Then, substituting these values into (x/y)^2, we get 4.

The total volume of infinitely many cubes with side lengths decreasing according to the sequence 1, 1/√2, 1/√3, ... is finite, while the total surface area is infinite.

The expression 2^(n+4) - 2(2^n) / 2(2^(n+3)) simplifies to 7/8.

The derivative of xy + sin(x) = e^y with respect to x is found using implicit differentiation, applying the product rule and chain rule, and solving for dy/dx.

The derivative of x² + y² = 25 with respect to x is -x/y.

The derivative of sin(2x² + 3x) is found using the chain rule, differentiating the outer sine function and the inner quadratic function separately, and then multiplying the results: cos(2x² + 3x) * (4x + 3).