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.

Substituting x = -1 into the equation gives (-1)^2 + 2(-1) + k = 0, leading to 1 - 2 + k = 0, thus 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.

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

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

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

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

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.

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.

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

Original area = π(10)² = 314 cm²; New area = π(8)² = 201.06 cm². Change = 314 - 201.06 = 112.94 cm².

Area change = π[(r-2)² - r²] = π[-4r + 4] = 3.14 * (-4r + 4).

Area = πr²; if r is doubled, area = π(2r)² = 4πr², so it quadruples.