Other types of equations

Solve the following biquadratic equation using the substitution $x^2 = t$:

$x^4 - 10x^2 + 9 = 0$

Find all real solutions to the following incomplete biquadratic equation by taking out a common factor:

$2x^4 - 32x^2 = 0$

Solve the following biquadratic equation and state clearly which of its solutions are real and why the others are discarded:

$x^4 - 2x^2 - 8 = 0$

Solve the following rational equation by multiplying all terms by $x$ to clear the denominator:

$\frac{12}{x} = x - 4$

Find the value of $x$ that satisfies the following simple rational equation:

$\frac{x+3}{x-1} = 2$

Solve the following rational equation by finding a common denominator. Pay special attention to possible solutions that make any denominator zero:

$\frac{x}{x-2} + \frac{3}{x+2} = \frac{8}{x^2-4}$

Solve the radical equation by isolating the square root and squaring both sides:

$\sqrt{3x + 4} = 5$

Determine the values of $x$ that satisfy the following radical equation. Remember that you must check the solutions at the end to rule out any false solutions:

$\sqrt{x + 2} = x$

Solve the radical equation by isolating one of the square roots and squaring. Remember to check the solution obtained:

$\sqrt{x+4} - \sqrt{x-1} = 1$

Solve the following exponential equation by expressing both sides as powers of the same base:

$3^{2x-1} = 27$

Find the value of $x$ in the logarithmic equation by directly applying the definition of a logarithm:

$\log_2(x + 5) = 3$

The number of members on a Discord server follows the exponential formula $N = 500 \cdot 2^{t/3}$, where $t$ is the time elapsed in months. How many months must pass for the server to reach exactly $4000$ members?