Algebra

Factoring gives (ax - b)(ax - 2b) = 0, so the roots are b/a and 2b/a, assuming a and b are nonzero. Thus, a/b and b/a are not generally both roots.

Discriminant D = b^2 - 4ac = (-3)^2 - 4(2)(5) = 9 - 40 = -31. Since D < 0, the equation has no real roots.

Using the quadratic formula x = [-b +/- sqrt(b^2 - 4ac)] / 2a. For 2x^2 - 7x + 3 = 0, x = [7 +/- sqrt(49 - 24)] / 4 = [7 +/- 5] / 4. Roots are 12/4 = 3 and 2/4 = 1/2.

Expand the equation: 2x^2 + x - 6x - 3 = x^2 + 5x. Simplify: 2x^2 - 5x - 3 = x^2 + 5x. x^2 - 10x - 3 = 0. This is a quadratic equation.

For equal roots, D = b^2 - 4ac = 0. Here, (-2*sqrt(5))^2 - 4(k)(4) = 0. So 20 - 16k = 0, which means 16k = 20, or k = 20/16 = 5/4.

Expanding the right side: (2x + 5)(5x - 3) = 10x^2 - 6x + 25x - 15 = 10x^2 + 19x - 15. The equation becomes 16x^2 - 3 = 10x^2 + 19x - 15, which simplifies to 6x^2 - 19x + 12 = 0. Since this is in the form ax^2 + bx + c = 0, it is a quadratic equation.

The equation is 3x/4 - 5x^2/8 = 7/8. Multiplying by 8 gives 6x - 5x^2 = 7, or 5x^2 - 6x + 7 = 0. This is a quadratic equation because it is in the form ax^2 + bx + c = 0 with a != 0.

For equal roots, discriminant D = b^2 - 4ac = 0. k^2 - 4(2)(3) = 0. k^2 - 24 = 0. k^2 = 24. k = +/- sqrt(24) = +/- 2 * sqrt(6).

Expanding (x+2)^3 gives x^3 + 6x^2 + 12x + 8. The equation becomes x^3 + 6x^2 + 12x + 8 = x^3 - 4. Subtracting x^3 from both sides results in 6x^2 + 12x + 12 = 0, which is a quadratic equation.

For equal roots, the discriminant D = b^2 - 4ac = 0. Here, p^2 - 4(2)(8) = 0. p^2 - 64 = 0, so p^2 = 64, p = +/- 8.

For x^2 - bx + 1 = 0 to have no real roots, the discriminant D < 0. D = (-b)^2 - 4(1)(1) = b^2 - 4. b^2 - 4 < 0 => b^2 < 4 => -2 < b < 2.

The displayed equation contains p x^2, making its formatting ambiguous. Under the usual intended equation 3x^2 + px + 3 = 0, the roots have product 1, so if they are r and r^2, then r^3 = 1 and r = 1; their sum gives p = -6.

If roots are 4 and 5, the equation is (x-4)(x-5) = x^2 - 9x + 20 = 0.

Let the roots be r and r^2. By Vieta's formulas, r + r^2 = -k and r * r^2 = -8. Thus, r^3 = -8, which gives r = -2. Substituting r = -2 into the sum equation, -2 + (-2)^2 = -k, so -2 + 4 = -k, which means k = -2.