Quadratic functions

Solve the inequality $(3 - x)(5x + 8) \geq 9 - 3x$.

Find the values of $k$ for which the equation $4x^2 - k = 4kx - 2$ has no real roots.

DO NOT USE A CALCULATOR IN THIS QUESTION.

Solve the equation $(2 + \sqrt{5})x^2 = 4x + 3(2 - \sqrt{5})$, giving your answers in the form $a + b\sqrt{5}$ where $a$ and $b$ are integers.

Solve the equation

Give your answers in exact form.

DO NOT USE A CALCULATOR IN THIS QUESTION.

Find the positive solution of the equation $(5 + 4\sqrt{7})x^2 + (4 - 2\sqrt{7})x - 1 = 0$, giving your answer in the form $a + b\sqrt{7}$, where $a$ and $b$ are fractions in their simplest form.

The curve $y = 2x^2 + k + 4$ intersects the straight line $y = (k + 4)x$ at two distinct points. Find the possible values of $k$.

Find the values of $k$ for which the line $y = 1 - 2kx$ does not meet the curve $y = 9x^2 - (3k + 1)x + 5$.

Find the set of values of $k$ for which the equation $kx^2 + 3x - 4 + k = 0$ has no real roots.