Algebra

For the roots to be real, the discriminant must be >= 0. For ax^2 - 2bx + c = 0, D = 4b^2 - 4ac >= 0 => b^2 >= ac. For bx^2 - 2sqrt(ac)x + b = 0, D = 4ac - 4b^2 >= 0 => ac >= b^2. Both conditions imply ac = b^2.

x^2 + x = 2x - 2 => x^2 - x + 2 = 0. Discriminant D = b^2 - 4ac = (-1)^2 - 4(1)(2) = 1 - 8 = -7. Since D < 0, the roots are imaginary.

Roots are imaginary if the discriminant D = b^2 - 4ac < 0. For a > 0 and c > 0, 4ac is positive, making b^2 - 4ac < 0 if b is 0.

For non-real roots, the discriminant D < 0. D = [2(p-2)]^2 - 4(p-2)(2) = 4(p-2)^2 - 8(p-2) = 4(p-2)(p-2-2) = 4(p-2)(p-4). For D < 0, (p-2)(p-4) < 0, which occurs when 2 < p < 4.

For equal roots, discriminant D = 0. Eq 1: p^2 - 4(64) = 0 => p^2 = 256 => p = 16 (since p>0). Eq 2: (-8)^2 - 4(p) = 0 => 64 = 4p => p = 16. Both equations yield p=16.

Let roots be alpha and beta. alpha - beta = 1. (alpha - beta)^2 = 1. (alpha + beta)^2 - 4*alpha*beta = 1. (-p)^2 - 4q = 1. p^2 = 4q + 1.

Let roots be a and a^2. Sum of roots: a + a^2 = 30. Product of roots: a * a^2 = a^3 = p. From a^2 + a - 30 = 0, (a+6)(a-5) = 0. If a=5, a^3 = 125. If a=-6, a^3 = -216. Thus p = 125 or -216.

For ax^2 + x + b = 0, real roots imply discriminant D = 1 - 4ab > 0, so 4ab < 1. For the second equation x^2 - 4*sqrt(ab)x + 1 = 0, the discriminant is D' = (4*sqrt(ab))^2 - 4(1)(1) = 16ab - 4. Since 4ab < 1, then 16ab < 4, which means 16ab - 4 < 0. A negative discriminant indicates imaginary roots.

If the roots are x and -x, the equation 1/(x+p) + 1/(x+q) = 1/r becomes (2x + p + q) / (x^2 + x(p+q) + pq) = 1/r. Cross-multiplying gives r(2x + p + q) = x^2 + x(p+q) + pq. For the roots to be symmetric around zero, the coefficient of the x term must be zero, implying 2r = p + q, or r = (p+q)/2.

Roots r1, r2. r1-r2 = r1*r2. (r1-r2)^2 = (r1+r2)^2 - 4r1r2. So (r1*r2)^2 = (r1+r2)^2 - 4r1r2. Sum = (a+1)/2, Product = (a-1)/2. ((a-1)/2)^2 = ((a+1)/2)^2 - 4((a-1)/2). (a-1)^2/4 = (a+1)^2/4 - 2(a-1). (a^2-2a+1) = (a^2+2a+1) - 8a + 8. -2a = -6a + 8. 4a = 8, a=2.

For real roots, the discriminant D >= 0. D = (-8)^2 - 4(1)(a^2 - 6a) = 64 - 4a^2 + 24a >= 0. Dividing by -4: a^2 - 6a - 16 <= 0. Factors: (a-8)(a+2) <= 0. Thus, -2 <= a <= 8.

Equation: 2x^2 + (3k - 6 - 15)x + (4 - k) = 0, so 2x^2 + (3k - 21)x + (4 - k) = 0. If roots are negative of each other, their sum is 0. Thus 3k - 21 = 0, so k = 7.

For roots to have opposite signs, the product of roots (c/a) must be negative. Here, p(p - 1)/3 < 0. This inequality holds when p is between 0 and 1.

Let the roots be alpha and beta. alpha + beta = p and alpha * beta = 8. Given |alpha - beta| = 2. Using (alpha - beta)^2 = (alpha + beta)^2 - 4*alpha*beta, we get 2^2 = p^2 - 4(8), so 4 = p^2 - 32. Thus p^2 = 36, meaning p = +/- 6.