Setting the discriminant to zero: (2p)^2 - 4(1)(p^2 - 4) = 0 leads to p = ±2.

For equal roots, the discriminant must be zero: 2^2 - 4*1*k = 0, leading to k = 1.

For no real roots, the discriminant must be less than zero: 2^2 - 4*1*k < 0 => 4 - 4k < 0 => k > 1.

For no real roots, the discriminant must be less than zero: 2^2 - 4*1*k < 0, hence k > 1.

For equal roots, the discriminant must be zero: 2^2 - 4*1*k = 0 leads to k = -1.

If one root is -2, then substituting x = -2 gives: (-2)^2 + 4(-2) + c = 0 => 4 - 8 + c = 0 => c = 4.

Using the formula for roots, k = (-2)^2 - 4*(-2) = 4 + 8 = 12.

The discriminant is 0, indicating that the roots are real and equal.

Using Vieta's formulas, k = (-2)(-4) = 8.

For both roots to be negative, k must be greater than the square of half the coefficient of x, hence k > 9.

The sum of the roots is 3 + (-3) = 0, so b = -0.

For equal roots, the discriminant must be zero: k^2 - 4*1*16 = 0, thus k = -8.

The discriminant must be less than zero: k^2 - 4*1*9 < 0 => k^2 < 36 => |k| < 6.

Using Vieta's formulas, the product of the roots is n = 1 * (-3) = -3.

Using Vieta's formulas, m = -(1 + (-3)) = 2.

Using Vieta's formulas, m = -(-1) = 1 and n = 2*(-3) = -6, thus m + n = 1 - 6 = -5.

Using Vieta's formulas, p = -(2 + 3) = -5 and q = 2*3 = 6. Thus, p + q = -5 + 6 = 1.

The sum of the roots is -p = 2 + 3 = 5, so p = -5.

For equal roots, the discriminant must be zero: k^2 - 36 = 0, hence k = 6.

The electric field at the surface of a sphere is given by E = Q/(4πε₀R²). If R is halved, E increases by a factor of 4.

The magnetic field at the center of a circular loop is inversely proportional to the radius; thus, doubling the radius halves the magnetic field.

The magnetic field at the center of a circular loop is given by B = (μ₀I)/(2r). If the radius is doubled, the magnetic field strength is halved.

The magnetic field at the center is inversely proportional to the radius, so it quadruples.

The moment of inertia of a disc is I = 1/2 MR^2. If R is doubled, I becomes 1/2 M(2R)^2 = 2MR^2, which is quadrupled.

The moment of inertia of a disk is I = 1/2 MR^2. If R is doubled, I becomes 1/2 M(2R)^2 = 2MR^2, which is 4 times the original.

If the radius is halved, the gravitational acceleration becomes four times stronger, as g is inversely proportional to the square of the radius.

g ∝ 1/R². If R is halved, g becomes 4 times greater.

Angular momentum L = Iω; if radius is doubled, moment of inertia I increases by a factor of 4, hence L quadruples.

Linear velocity v = rω. If r is halved and ω remains constant, v also halves.

Moment of inertia I is proportional to the square of the radius, so halving the radius reduces I to one-fourth.