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The solution to the equation \sqrt{(X - 4)(X + 4)} = 3 is X = 5 and X = -5.

The integrals were evaluated using trigonometric substitution, partial fraction decomposition, and substitution. Integral c) was solved using trigonometric substitution, resulting in x/(√(1-x^2)) + C. Integral d) diverged to infinity. Integral e) was solved using partial fraction decomposition, leading to a solution involving natural logarithms and a constant term.

The solution involves finding the values of a and b, given a*a = 8 and a*b = 24. Then, b*b is calculated, resulting in a final answer of 72.

Cramer's rule is used to solve a system of four linear equations with four unknowns, resulting in the values of x1, x2, x3, and x4.

The equation of the line equidistant from the parallel lines y = (1/3)x - 4 and y = (1/3)x + 1 is y = (1/3)x - 5/3.

The solution finds the equation of a line g' that is the reflection of line g across line s. It involves finding the intersection point of g and s, projecting a direction vector of g onto a normal vector of s, and then calculating the direction vector of g'. Finally, the equation of g' is determined using the intersection point and the direction vector of g'.

The square root of 19 squared plus 19 plus 20 is calculated to be 20.

To find (a+b)^2, given a^2 + b^2 = 29 and ab = 10, substitute the given values into the equation (a+b)^2 = a^2 + 2ab + b^2, resulting in (a+b)^2 = 49.

The expression 3³ ÷ 3 ÷ 3 simplifies to 1.

The only real solution to the cubic equation a^3 + a^2 - 36 = 0 is a = 3.