Exercises: Other equations

Solve the following biquadratic equation, 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 to 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 equation with absolute value:

$|2x - 5| = 7$

Factorise 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 $10$ cm and its area is $48\text{ cm}^2$. Set up a biquadratic equation to find the lengths 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 bacterial culture grows according to an exponential law. To determine the critical moment of an experiment, a biologist must solve the equation $4^x - 5 \times 2^x + 4 = 0$. Find the values of $x$ by applying a substitution.

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 real values of $x$ satisfying the following equation with quadratic modulus:

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

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.

A sample of radioactive material decays such that the remaining amount $M(t)$ in grams after $t$ years obeys the expression $M(t) = 200 \times e^{-0.05t}$. How many years must pass for exactly $50$ grams 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. Give full justification for your answer.

A cyclist travels from town A to town B, separated by $60$ km, at a constant speed. On the return journey, due to a headwind, their speed decreases by $5$ km/h, taking $1$ hour longer to complete the same journey. Set up the rational equation that models the situation and determine the speed on the outward journey.

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

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

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

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 real solutions of the equation with nested modulus:

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

A card's PIN consists of four positive, consecutive integers in increasing order. If we construct a monic polynomial $P(x)$ of degree $4$ whose roots are precisely those four digits, its constant term turns out to be equal to $24$. Determine the expanded polynomial $P(x)$ and the four-digit PIN.

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.