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

If α, β, γ  and \(\delta\) are the roots of the polynomial equation 2x⁴ + 5x³ − 7x² + 8 = 0, find a quadratic equation with integer coefficients whose roots are α + β + γ + \(\delta\) and αβ૪\(\delta\).

Find a polynomial equation of minimum degree with rational coefficients, having \(\sqrt{5}\)−\(\sqrt{3}\) as a root.

Solve: (2x-1) (x+3) (x-2) (2x+3)+20 = 0

Solve the equation 3x³-26x²+52x - 24 = 0 if its roots form a geometric progression.

Determine k and solve the equation 2x³-6x²+3x+k = 0 if one of its roots is twice the sum of the other two roots.

Solve the equation : x⁴-14x² + 45 = 0

Solve the cubic equations: 8x³ - 2x² - 7x + 3 = 0

If sin ∝, cos ∝ are the roots of the equation ax² + bx + c-0 (c ≠ 0), then prove that (n + c)² - b² + c²

Find the condition that the roots of cubic x³+ ax²+ bx + c = 0 are in the ratio p : q : r.

If p is real, discuss the nature of the roots of the equation 4x²+ 4px + p + 2 = 0 in terms of p.

If 2+i and 3-\(\sqrt{2}\) are roots of the equation x⁶-13x⁵+ 62x⁴-126x³+ 65x²+127x-140 = 0, find all roots.

Find the condition that the roots of ax³+ bx²+ cx + d = 0 are in geometric progression. Assume a, b, c, d ≠ 0.