Other equations

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

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

The diagonal of a rectangle measures $10$ cm and its area is $48\text{ cm}^2$. Set up a biquadratic equation to find the lengths of its sides and solve it.

Determine the value of $x$ that satisfies the following exponential equation:

$3^{2x+1} - 27 = 0$

A bacterial culture grows according to an exponential law. To determine the critical moment of an experiment, a biologist must solve the equation $4^x - 5 \cdot 2^x + 4 = 0$. Find the values of $x$ by applying a substitution.

Apply the properties of logarithms to solve the equation:

$\log_2(x+3) + \log_2(x-3) = 4$

Apply the properties of logarithms to group the terms and solve the equation:

$\log(x+1) - \log(x-1) = \log(x+4) - \log(x+1)$

Factorise the polynomial and solve the cubic equation:

$x^3 - 4x^2 - x + 4 = 0$

An open-topped box is to be constructed from a rectangular piece of cardboard measuring $20 \times 15$ cm, by cutting out equal squares of side length $x$ from the corners and folding up the sides. If the volume of the box must be $378\text{ cm}^3$, set up the corresponding cubic polynomial equation and find the value of $x$, knowing that it is an integer.

Determine for which values of the parameter $k$ the biquadratic equation $x^4 - kx^2 + 9 = 0$ has exactly two distinct real solutions. Give full justification for your answer.

Solve the following exponential equation involving different bases by taking out common factors on each side:

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

A sample of radioactive material decays such that the remaining amount $M(t)$ in grams after $t$ years obeys the expression $M(t) = 200 \cdot e^{-0,05t}$. How many years must pass for exactly $50$ grams of material to remain? Express the exact result using natural logarithms.

The energy $E$ (in ergs) released by an earthquake is related to its magnitude $M$ on the Richter scale by the equation $\log E = 11,8 + 1,5M$. If an earthquake releases $1000$ times more energy than another whose magnitude is $4,2$, what is the magnitude of the first earthquake on the Richter scale?

Solve the following non-linear system modelling the intersection of two paths in the Cartesian plane: $y = \sqrt{x+2}$ and $y = |x|$. Determine the points of intersection algebraically.