If x=\(\sqrt { 2 } +\sqrt { 3 } \)  find \(\frac { { x }^{ 2 }+1 }{ { x }^{ 2 }-2 } \)

Prove that \(log_{10}2+16log_{10}\frac { 16 }{ 15 } +12log_{10}\frac { 25 }{ 24 } +7log_{10}\frac { 81 }{ 80 } =1\)

Find the condition that one of the roots of ax²+ bx + c may be negative of the other.

A manufacturer has 600 litres of a 12 percent solution of acid. How many litres of a 30 percent acid solution must be added to it so that the acid content in the resulting mixture will be more than 15 percent but less than 18 percent?

Find all values of x for which \({{x^3(x-1)}\over{x-2}}>0.\)

Solve : \({{x^2-4}\over{x^2-2x-15}}\le0\)

Resolve the following rational expressions into partial fractions.
\({{x}\over{{(x-1)}^{3}}}\)

Resolve the following rational expressions into partial fractions.
\({{1}\over{x^4-1}}\)

Resolve the following rational expressions into partial fractions.
\({{x^2+x+1}\over{x^2-5x+6}}\)

Determine the region in the plane determined by the inequalities.
\(x-2y\ge 0,\ 2x-y\le -2,\ x\ge 0,\ y\ge 0.\)