Other types of equations

Solve the following rational equation, paying attention to possible extraneous solutions that make the denominators zero:

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

A cyclist travels a distance of $60\text{ km}$ at a constant speed $v$. If they had ridden $5\text{ km/h}$ faster, it would have taken them $1$ hour less to complete the same journey. Knowing that the time taken is $t = \frac{d}{v}$, set up a rational equation and determine the initial speed of the cyclist.

Find the value or values of $x$ that satisfy the following radical equation. Remember to check the validity of the solutions obtained:

$\sqrt{2x+5} - 3 = x$

Isolate the radical, square both sides and solve the irrational equation. Check whether the solutions found are valid:

$\sqrt{x^2+5x+1} = x+2$

Solve the following irrational equation involving two radicals. Isolate one of them, square both sides, and repeat the process if necessary. Do not forget to check the validity of your results in the original equation:

$\sqrt{x+4} + \sqrt{x-1} = 5$

Solve the following biquadratic equation, finding all its real solutions:

$x^4 - 13x^2 + 36 = 0$

Solve the following incomplete biquadratic equation by taking out a common factor:

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