Find \(A \) and \(B \), if \(\dfrac{x}{{x + 1}} – \dfrac{{x – 1}}{{x – 2}} = \dfrac{{Ax + B}}{{\left( {x + 1} \right)\left( {x – 2} \right)}}\) in its simplest form of a fraction.

Find \(A \), if \(\dfrac{{14}}{{\left( {x – 8} \right)\left( {2x – 3} \right)}} \div \dfrac{{x – 7}}{{2\left( {x – 8} \right)}} = \dfrac{A}{{\left( {2x – 3} \right)\left( {x – 7} \right)}}\).

Find \(A\), \(B\)  and  \(C\)  if  \({\left( {x – 5} \right)^2} = A{x^2} + Bx + C\).

Find  \(d\),   given  \(f = \dfrac{{{a^2} – {d^2}}}{{4a}}\)  where  \(f = 2\),  and \(a = 15\).

Sean is biking with a constant speed towards town B from town A.   With the aid of wind, which has a speed of 2.5 km it takes him only one hour to reach the destination.   However, on the way back to town A he has to bike against the wind, and it takes him three hours to reach town A.   Find Sean’s speed.

Find \(A \), \(B \) and \(C \), if \(\dfrac{{12{x^3} \times 4{y^8}}}{{4x \times y}} = A{x^B}{y^C}\).

Find \( x \), if \({\left( {{4^{ – 2}}} \right)^{ – 1}} = {2^x}\).

Simplify \(2x{\left( {{x^2} – {y^2}} \right)^{ – 1}} – {\left( {x – y} \right)^{ – 1}}\).

Solve   \({4^x} – {2^x} – 2 = 0\).

A sequence is defined by \({u_n} = 2 \times {3^{n – 1}}\). Find \({u_3}\).