Find the sum of squares of roots of the equation 2x⁴- 8x³+ 6x²-3 = 0.

If the equations x² + px + q = 0 and x² + p'x + q' = 0 have a common root, show that it must  be equal to \(\frac { pq'-p'q }{ q-q' } \) or \(\frac { q-q' }{ p'-p } \).

A 12 metre tall tree was broken into two parts. It was found that the height of the part which was left standing was the cube root of the length of the part that was cut away. Formulate this into a mathematical problem to find the height of the part which was left standing.

If α and β are the roots of the quadratic equation 17x²+43x−73 = 0 , construct a quadratic equation whose roots are α + 2 and β + 2.

If α and β are the roots of the quadratic equation 2x²−7x+13 = 0 , construct a quadratic equation whose roots are α² and β².

If α, β, and γ are the roots of the equation x³ + px² + qx + r = 0, find the value of  \(\Sigma \frac { 1 }{ \beta \gamma } \) in terms of the coefficients.

Find the monic polynomial equation of minimum degree with real coefficients having 2 -\(\sqrt{3}\)i as a root.

Form a polynomial equation with integer coefficients with \(\sqrt { \frac { \sqrt { 2 } }{ \sqrt { 3 } } } \) as a root.

Prove that a line cannot intersect a circle at more than two points.