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

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

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.

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.

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 following rational equation by finding a common denominator. Pay special attention to possible solutions that make any denominator zero:

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

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 $x$.

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.

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 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 radical equation by isolating one of the roots and squaring. Remember to check the solution obtained:

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

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

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

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.

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.

Solve the following exponential equation by expressing both sides as powers of the same base:

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

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 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 $u = 2^t$).

Find the value of $x$ in the logarithmic equation by applying the definition of a logarithm directly:

$\log_2(x + 5) = 3$

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

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

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.