Exercises: Other types of equations

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

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

Find all the real solutions of the following incomplete biquadratic equation by factoring out the common factor:

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

Solve the following rational equation by multiplying all the terms by $x$ to eliminate 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 radical equation by isolating the radical 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 possible extraneous solutions:

$\sqrt{x + 2} = x$

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 applying the definition of a logarithm directly:

$\log_2(x + 5) = 3$

The area of a rectangle is $6 \text{ cm}^2$. Knowing that its base measures $x^2 + 2$ in. and its height $x^2 - 3$ in., set up a biquadratic equation to find the value of $x$ (with $x > 0$) and determine the real dimensions of the rectangle.

A cyclist travels to a neighboring town at a speed of $20 \text{ km/h}$ and returns along the same route at a speed of $x \text{ km/h}$. The average speed for the whole round trip is given by the formula $v_m = \frac{2 \cdot 20 \cdot x}{20 + x}$. If the total average speed was $24 \text{ km/h}$, set up a rational equation and calculate the return speed $x$.

The period $T$ of oscillation of a simple pendulum (in seconds) is given by the formula $T = 2\pi \sqrt{\frac{L}{g}}$, where $L$ is the length of the string in feet and $g$ is the acceleration due to gravity. Taking the approximation $\pi^2 \approx 10$ and $g = 10 \text{ ft/s}^2$, find the length $L$ of the pendulum so that its period is exactly $2$ seconds.

The number of bacteria in a laboratory culture follows the exponential formula $N = 500 \cdot 2^{t/3}$, where $t$ is the time elapsed in hours. How many hours must elapse for the population to reach exactly $4000$ bacteria?

The sound intensity level $L$ in decibels (dB) is calculated using the logarithmic expression $L = 10 \log\left(\frac{I}{10^{-12}}\right)$, where $I$ is the acoustic intensity in $\text{W/m}^2$. Calculate the intensity $I$ of a sound produced by heavy traffic, knowing that its level is $80 \text{ dB}$.

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

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

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 one of the roots and squaring. Remember to check the solution obtained:

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

The diagonal of a rectangular wooden frame measures $\sqrt{10}$ in. If one of its sides measures $x$ in. and the other measures $\frac{3}{x}$ in., set up a rational equation, transform it into a biquadratic equation, and calculate the dimensions of the frame.

A water tank for irrigation can be filled using two different faucets. The first faucet takes $x$ hours to fill it on its own, and the second faucet, which has a lower flow rate, takes $x+3$ hours. If both are opened at the same time, the tank fills in exactly $2$ hours. Set up a rational equation and determine how long each faucet takes separately.

In a piece of sheet metal shaped like a right triangle, one leg measures $x$ in., the other leg measures $\sqrt{x+5}$ in., and the hypotenuse measures $x+1$ in. Set up a radical equation by applying the Pythagorean theorem, find the value of $x$, and calculate the area of the piece.

The value of a financial investment (in thousands of dollars) follows the evolution $V(t) = 4^t - 6 \cdot 2^t + 10$, where $t$ is the time in years since the start of the investment. At what times $t$ is the value of the investment exactly $2$ thousand dollars? (Hint: use the properties of exponents and make the substitution $u = 2^t$).

The magnitude of an earthquake on the Richter scale is given by $M = \log(A) - \log(A_0)$, where $A$ is the maximum amplitude of the recorded seismic waves and $A_0$ is a standard reference amplitude. If in an earthquake of magnitude $M_1 = 3$ the amplitude was $A_1$, and in another stronger earthquake the amplitude was $A_2 = 100 \cdot A_1$, what was the magnitude $M_2$ of this second earthquake? Apply the properties of logarithms to prove it step by step.

A group of friends rents a cabin for the weekend for a total of $\$120$. At the last minute, $2$ more friends sign up for the trip, which means that each of those who were initially going pays $\$3$ less. If $x$ is the initial number of friends, set up a rational equation and find out how many friends finally went to the cabin.

Solve the following radical equation involving two radicals. You will have to square twice. Check the solutions obtained at the end of the process:

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

Solve the following logarithmic equation by applying the properties of logarithms (sum of logarithms) and check the result to rule out solutions that generate negative arguments:

$\log(x+2) + \log(x-1) = 1$