Algebra

If the roots are p and q, then p+q = -b/a and pq = c/a. The sum of reciprocals is (1/p) + (1/q) = (p+q)/(pq) = (-b/a) / (c/a) = -b/c.

For ax^2 + bx + c = 0, the sum of roots p + q = -b/a and product pq = c/a. The expression p/q + q/p = (p^2 + q^2) / pq = ((p + q)^2 - 2pq) / pq. Substituting the values: ((-b/a)^2 - 2(c/a)) / (c/a) = (b^2/a^2 - 2c/a) / (c/a) = (b^2 - 2ac) / ac.

Sum of roots = -(2k+1)/3. Product of roots = -5/3. Equating them: -(2k+1)/3 = -5/3 => 2k+1 = 5 => 2k = 4 => k = 2.

For real roots, discriminant D = b^2 - 4ac >= 0. 3^2 - 4(5)(-p) >= 0. 9 + 20p >= 0. 20p >= -9, so p >= -9/20.

For a quadratic equation ax^2 + bx + c = 0 to have real roots, the discriminant D = b^2 - 4ac must be >= 0. Here, 8^2 - 4(5)(p) >= 0, which means 64 - 20p >= 0. Thus, 64 >= 20p, or p <= 64/20 = 16/5.

For distinct roots, the discriminant D > 0. D = (-2a)^2 - 4(2)(4) = 4a^2 - 32. 4a^2 - 32 > 0 implies a^2 > 8. Thus |a| > sqrt(8) = 2*sqrt(2).

For real and distinct roots, the discriminant D = b^2 - 4ac > 0. Here, 8^2 - 4*1*p > 0, so 64 - 4p > 0, which means 4p < 64, or p < 16.

For x^2 + px + 64 = 0, D = p^2 - 256 >= 0 implies |p| >= 16. For x^2 - 8x + p = 0, D = 64 - 4p >= 0 implies p <= 16. Both conditions are satisfied only when p = 16 or p = -16. Since 16 is an option, it is the correct value.

Expanding (3x+1)(x-1)=15 gives 3x^2 - 3x + x - 1 = 15, which simplifies to 3x^2 - 2x - 16 = 0. The sum of the roots of a quadratic equation ax^2 + bx + c = 0 is -b/a. Here, the sum is -(-2)/3 = 2/3.

For a quadratic equation with complex coefficients, the roots do not necessarily come in conjugate pairs. Using the sum of roots: (-1+i) + r2 = -(1-3i) = -1+3i. r2 = -1+3i - (-1+i) = 2i.

Let |x| = y. The equation becomes y^2 - 3y + 2 = 0, which factors to (y-1)(y-2) = 0. So |x| = 1 or |x| = 2. This gives x = 1, -1, 2, -2, totaling four real roots.