Exercises: Other equations

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

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

Find the value of $x$ that satisfies the following equation with the unknown in the denominator:

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

Find the real solutions of the irrational equation, checking their validity:

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

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

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

Apply the properties of logarithms to solve the equation:

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

Find the values of $x$ that satisfy the modulus equation:

$|2x - 5| = 7$

Factor the polynomial and solve the cubic equation:

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

Solve the following rational equation by reducing it to a polynomial equation:

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

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

Two hoses fill a tank in $6$ hours working together. If the first hose takes $5$ hours less than the second to fill the tank on its own, how long does each one take? Set up and solve the corresponding rational equation.

Solve the following irrational equation with two radicals and check the solution obtained:

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

A bacteria 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 change of variable.

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

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

Find all the real values of $x$ that satisfy the following quadratic modulus equation:

$|x^2 - 5x + 4| = 2$

A box without a lid is to be made from a rectangular piece of cardboard of $20 \times 15$ in, by cutting out equal squares of side $x$ from the corners and folding up the edges. 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.

A sample of radioactive material decays such that the remaining amount $M(t)$ in ounces after $t$ years obeys the expression $M(t) = 200 \cdot e^{-0.05t}$. How many years must pass for exactly $50$ ounces of material to remain? Express the exact result using natural logarithms.

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

A cyclist travels from town A to town B, separated by $60$ miles, at a constant speed. On the return journey, due to a headwind, their speed decreases by $5$ mph, so it takes $1$ hour more to complete the same route. Set up the rational equation that models the situation and determine the outward speed.

A telecommunications tower is located on the straight bank of a river. A cable is to be laid in a straight line from a point $P$ on the bank to the tower, and another cable from $P$ to a cabin located $5$ miles downstream from the tower and $1$ miles inland. If the total length of the cable used is $6$ miles, set up an irrational equation and determine at what distance from the tower point $P$ is located.

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

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

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?

Find all the real solutions of the nested modulus equation:

$||x - 2| - 3| = 1$

A card PIN number is made up of four positive, consecutive integer digits in increasing order. If we form a monic polynomial $P(x)$ of degree $4$ whose roots are exactly those four digits, its constant term is equal to $24$. Determine the expanded polynomial $P(x)$ and the four-digit PIN number.

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