Algebra

(x-1)^2 = 4/9. Taking the square root of both sides: x-1 = +/- 2/3. x = 1 + 2/3 = 5/3 or x = 1 - 2/3 = 1/3.

For real roots, discriminant D >= 0. D = k^2 - 4(1)(4) = k^2 - 16. k^2 - 16 >= 0 => k^2 >= 16. So k >= 4 or k <= -4. The least positive value is 4.

Discriminant D = a^2 - 4b. Given a = 2*sqrt(b), D = (2*sqrt(b))^2 - 4b = 4b - 4b = 0. When the discriminant is 0, the roots are real and equal.

For equal roots, the discriminant D = b^2 - 4ac = 0. (2k+6)^2 - 4(k)(16) = 0. 4k^2 + 24k + 36 - 64k = 0. 4k^2 - 40k + 36 = 0. k^2 - 10k + 9 = 0. (k-1)(k-9) = 0. So k = 1 or 9.

Product of roots = c/a = 19/6. Sum of roots = -b/a = 12/6 = 2. Difference = 19/6 - 2 = 19/6 - 12/6 = 7/6.

For the equation 2x^2 + 3x + 7 = 0, the sum of roots alpha + beta = -3/2 and the product alpha * beta = 7/2. The expression 1/alpha + 1/beta = (alpha + beta) / (alpha * beta) = (-3/2) / (7/2) = -3/7.

For roots to be imaginary, the discriminant (b^2 - 4ac) must be less than 0. Here, 3^2 - 4(1)(2c) < 0. 9 - 8c < 0. 8c > 9, so c > 1.125. The least integer c is 2.

Expanding the equation: 4x^3 + 4x^2 + 12x + 5x^3 - 25x^2 + 15x = -1.5, which simplifies to 9x^3 - 21x^2 + 27x + 1.5 = 0. Dividing by 1.5 gives 6x^3 - 14x^2 + 18x + 1 = 0. By analyzing the derivative, the function is strictly increasing, meaning it has only one real root, hence two imaginary roots.

ax^2 + bx = 0. Factor out x: x(ax + b) = 0. Roots are x = 0 and ax + b = 0, so x = -b/a.

For a quadratic equation ax^2 + bx + c = 0 to have equal roots, the discriminant D = b^2 - 4ac must be zero. Here, q^2 - 4pr = 0, so q^2 = 4pr.

For a quadratic equation with rational coefficients, if the discriminant D is equal to 0, then the roots are real, rational, and equal. Therefore, statements P (roots are equal) and R (roots are rational numbers) are correct.

If y = 2/3 is a root, then 3(2/3)^2 - k(2/3) + 8 = 0. This is 3(4/9) - 2k/3 + 8 = 0, which is 4/3 - 2k/3 + 8 = 0. Multiplying by 3: 4 - 2k + 24 = 0, so 28 = 2k, k = 14.

x^2 - 4 = 0 implies x^2 = 4. Taking the square root of both sides gives x = 2 or x = -2.

x/k = k/x => x^2 = k^2 => x = k or x = -k.