Algebra

Let roots be r1, r2. r1+r2 = a, r1*r2 = b. |r1-r2| < c => (r1-r2)^2 < c^2 => (r1+r2)^2 - 4r1r2 < c^2 => a^2 - 4b < c^2 => 4b > a^2 - c^2 => b > (a^2 - c^2)/4. Also, for real roots, D >= 0 => a^2 - 4b >= 0 => b <= a^2/4.

If roots are alpha and alpha + p, then their sum is -b/a and product is c/a. (alpha + p) - alpha = p. The difference of roots is sqrt(D)/|a| = p. Thus, D/a^2 = p^2, which means (b^2 - 4ac) / a^2 = p^2, or b^2 - 4ac = a^2 * p^2.

If roots are in ratio 3:4, let them be 3k and 4k. Sum of roots = 7k = -q/p. Product of roots = 12k^2 = r/p. From the sum, k = -q/(7p). Substitute into the product: 12 * (-q/7p)^2 = r/p. 12 * (q^2 / 49p^2) = r/p. 12q^2 = 49pr.

Interpreting the garbled condition as no two of a, b, and c being equal, the determinant condition reduces to 2a + b + c = 0. Substituting this relation into the quadratic shows that at least one root lies between 0 and 1/2.

alpha + beta = 4, alpha * beta = 2. alpha^2 + beta^2 = (alpha + beta)^2 - 2(alpha * beta) = 4^2 - 2(2) = 16 - 4 = 12.

Set 49x^2 - 81 = 0. This is a difference of squares: (7x - 9)(7x + 9) = 0. The solutions are x = 9/7 and x = -9/7.

The polynomial is 2s^2 - (1 + 2*sqrt(2))s + sqrt(2). Factoring gives (2s - 1)(s - sqrt(2)) = 0. The zeroes are s = 1/2 and s = sqrt(2).

Let x1 = sec^2(pi/7) and x2 = tan^2(pi/7). We know sec^2(theta) - tan^2(theta) = 1. The roots of ax^2 + bx + c = 0 satisfy x1 + x2 = -b/a and x1*x2 = c/a. Substituting these into the expression leads to the value 1.

Sum of roots = 7. Product of roots = 2. Difference = 7 - 2 = 5.

For a quadratic equation ax^2 + bx + c = 0, the sum of roots is -b/a. Here, -(-z)/8 = 10, so z/8 = 10, z = 80.

For a quadratic equation ax^2 + bx + c = 0, the product of roots is c/a. Here, c = -18 and a = 6. Product = -18 / 6 = -3.

For a quadratic equation ax^2 + bx + c = 0, the sum of roots is -b/a. Here, -20 / p = 20. Solving for p gives p = -1.