Algebra and Surds
JUPEB Mathematics

Solve the equation $2^{2x} - 5(2^x) + 4 = 0$.

Find the remainder when $x^3 - 4x^2 + x + 6$ is divided by $(x - 1)$.

Express $\dfrac{5x + 1}{(x - 1)(x + 2)}$ in partial fractions.

If $\alpha$ and $\beta$ are the roots of $2x^2 - 3x + 1 = 0$, find the value of $\alpha^2 + \beta^2$.

Solve $\log_2 (x - 1) + \log_2 (x + 1) = 3$.

Expand and simplify $(2 + \sqrt{5})^2$.

Solve the equation $\sqrt{x + 5} = x - 1$.

Evaluate $27^{\frac{2}{3}} \times 9^{-\frac{1}{2}}$.

If $f(x) = \dfrac{2x + 1}{x - 3}$, $x \neq 3$, find $f^{-1}(x)$.

Given that $x^2 - 1$ is a factor of $x^3 + ax^2 + bx - 2$, find the values of $a$ and $b$.

Simplify $\sqrt{50} + \sqrt{18} - \sqrt{8}$.

Rationalise the denominator of $\dfrac{3}{2 - \sqrt{3}}$.