Quadratic equation

The minimum possible value of the sum of the squares of the roots of the equation
x² + (a + 3)x - (a + 5) = 0 is

If u² + (u−2v−1)² = −4v(u + v), then what is the value of u + 3v?

The product of the distinct roots of |x² – x – 6| = x + 2 is

The quadratic equation x² + bx + c = 0 has two roots 4a and 3a, where a is an integer. Which of the following is a possible value of b² + c ?

Let m and n be positive integers, If x² + mx + 2n = 0 and x² + 2nx + m = 0 have real roots, then the smallest possible value of m + n is

Let a and b be natural numbers. If a² + ab + a = 14 and b² + ab + b = 28, then (2a + b) equals

The number of integer solutions of the equation (x² – 10)^{(x2 – 3x – 10)} = 1 is

Suppose k is any integer such that the equation 2x² + kx + 5 = 0 has no real roots and the equation x² + (k – 5) x + 1 = 0 has two distinct real roots for x. Then, the number of possible values of k is

If x and y are real numbers such that x² + (x – 2y – 1)² = –4y(x + y), then the value x – 2y is

Let α and β be the two distinct roots of the equation 2x² – 6x + k = 0, such that (α + β) and αβ are the distinct roots of the equation x² + px + p = 0. Then, the value of 8(k – p) is

A quadratic equation x² + bx + c = 0 has two real roots. If the difference between the reciprocals of the roots is 1/3, and the sum of the reciprocals of the squares of the roots is 5/9, then the largest possible value of (b + c) is