Algebra

Substitute x = 2 into 3x^2 - 9x = kx - k. 3(4) - 9(2) = k(2) - k. 12 - 18 = k. k = -6.

The equation is 4sqrt(3)x^2 + 5x - 2sqrt(3) = 0. Roots are given by quadratic formula: x = [-5 +/- sqrt(25 - 4(4sqrt(3))(-2sqrt(3)))] / 8sqrt(3) = [-5 +/- sqrt(25 + 96)] / 8sqrt(3) = [-5 +/- 11] / 8sqrt(3). Roots are 6 / 8sqrt(3) = 3 / 4sqrt(3) = sqrt(3)/4 and -16 / 8sqrt(3) = -2/sqrt(3).

The roots of x^2 - (sqrt(2)+1)x + sqrt(2) = 0 are found by factoring as (x - sqrt(2))(x - 1) = 0. Thus, the roots are sqrt(2) and 1.

Expanding ax^2 + (4a^2 - 3b)x - 12ab = 0 gives ax^2 + 4a^2x - 3bx - 12ab = 0. Factoring: ax(x + 4a) - 3b(x + 4a) = 0, which is (ax - 3b)(x + 4a) = 0. The roots are x = 3b/a and x = -4a.

The discriminant D = (8ab)^2 - 4(3a^2)(4b^2) = 64a^2b^2 - 48a^2b^2 = 16a^2b^2. Since a, b are not 0, D > 0, so the roots are real and distinct.

Rearranging gives x^2 - (2a + b)x + 2ab = 0. This factors as (x - 2a)(x - b) = 0, so the roots are 2a and b, not -2a and b. Therefore the statement is false.

The discriminant is (24abcd)^2 - 4(9a^2b^2)(16c^2d^2) = 576a^2b^2c^2d^2 - 576a^2b^2c^2d^2 = 0. A discriminant of 0 means the roots are real and equal.

Interpreting the garbled middle term as -25X, the sum of the roots is 25/5 = 5. The only pair of prime roots with sum 5 is 2 and 3, whose product is 6. Thus a^2 - a = 30, giving a = 6 or -5, and their sum is 1.

Squaring both sides gives 3y + 1 = y - 1, which leads to 2y = -2, so y = -1. Checking y = -1 in the original equation: sqrt(3(-1)+1) = sqrt(-2), which is undefined in real numbers. Thus, there are no real solutions.

If roots are equal in magnitude and opposite in sign, they are a and -a. The sum of these two roots is 0. Testing the options, if 1 and -1 are roots, (x-1)(x+1) = x^2-1 must be a factor. Dividing the polynomial by x^2-1 yields x^2+5x+6 = (x+2)(x+3). The roots are 1, -1, -2, -3.

Let y = (x - 1)/(x + 1). Solving for x gives x = (1 + y)/(1 - y). Substituting this into the original equation (x^2 - 2x + 3 = 0) results in ((1 + y)/(1 - y))^2 - 2((1 + y)/(1 - y)) + 3 = 0. Simplifying this leads to 3y^2 - 2y + 1 = 0.

Let the new roots be y = x + 2, so x = y - 2. Substitute this into the original equation: (y-2)^2 - 2(y-2) + 3 = 0. Expanding gives y^2 - 4y + 4 - 2y + 4 + 3 = 0, which simplifies to y^2 - 6y + 11 = 0.