For both roots to be positive, k must be greater than 9 (since the sum of roots is -b/a and must be positive).

For both roots to be positive, k must be greater than 12 (from Vieta's formulas). The minimum integer k is 13.

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.

For real roots, the discriminant must be non-negative: k^2 - 4*1*16 >= 0, leading to k^2 >= 64.

The discriminant must be positive: k^2 - 4*1*16 > 0, which simplifies to k^2 > 64, hence k < -8 or k > 8.

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

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

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

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.

Using Vieta's formulas, p = -(3 + (-2)) = -1 and q = 3 * (-2) = -6. Therefore, p + q = -1 - 6 = -7.

Using the sum of roots formula, p = -(3 + (-2)) = -1. Therefore, p = 1.

Using Vieta's formulas, p = -(3 + 4) = -7 and q = 3 * 4 = 12. Therefore, p + q = -7 + 12 = 5.

The equation can be factored as (x - 5)^2 = 0, giving the root x = 5.

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

Factoring gives (x - 3)(x - 5) = 0, hence the roots are 3 and 5.

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

The product of the roots gives k = (-2)(-3) = 6.

For the equation to have one real root, the discriminant must be zero. Thus, k must equal 4.

Factoring the equation gives (x - 2)(x - 3) = 0, thus the roots are 2 and 3.

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.

Using Pythagoras theorem: chord length = 2√(r² - d²) = 2√(4² - 3²) = 2√(16 - 9) = 2√7 ≈ 5.29 cm.

Circumference = 2πr = 2π × 7 cm = 14π cm.

Circumference = 2πr = 2 × (22/7) × 7 = 44 cm.

Circumference = 2πr = 2 × 3.14 × 7 cm = 43.96 cm.

Using the tangent length formula: length = √(distance from center² - radius²) = √(10² - 7²) = √(100 - 49) = √51, which is approximately 7.14.

Circumference = 2 * π * radius = 2 * π * 7 = 14π units.

Using the tangent length formula: length = √(distance from center² - radius²) = √(10² - 7²) = √(100 - 49) = √51 ≈ 7.14.