Exercises: Other types of equations

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

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

Find all real solutions to the following incomplete biquadratic equation by taking out a common factor:

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

Solve the following rational equation by multiplying all terms by $x$ to clear 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 square root 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 any false 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 directly applying the definition of a logarithm:

$\log_2(x + 5) = 3$

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

A cyclist travels to a neighbouring 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 $v_m = \frac{2 \times 20 \times x}{20 + x}$. If the overall average speed was $24 \text{ km/h}$, set up a rational equation and calculate the return speed $x$.

The distance $d$ (in meters) traveled by a car from the moment the brakes are applied until it stops can be approximated by the formula $d = \frac{v^2}{20}$, where $v$ is the speed in m/s. If a car leaves skid marks of $45$ m, how fast was it traveling?

The number of members on a Discord server follows the exponential formula $N = 500 \times 2^{t/3}$, where $t$ is the time elapsed in months. How many months must pass for the server to reach exactly $4000$ members?

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 sound 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 of its solutions are real 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 square roots and squaring. Remember to check the solution obtained:

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

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

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

In a piece of sheet metal shaped like a right-angled triangle, one leg measures $x$ cm, the other leg measures $\sqrt{x+5}$ cm and the hypotenuse measures $x+1$ cm. Formulate a radical equation applying Pythagoras' theorem, find the value of $x$ and calculate the area of the piece.

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

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 $A_2 = 100 \times A_1$, what was the magnitude $M_2$ of this second earthquake? Apply log rules to show it step by step.

A group of friends rents a vacation home for the weekend for a total of $2\,400$ pesos. At the last minute, $2$ more friends join the trip, which means each person who was originally going pays $60$ pesos less. If $x$ is the initial number of friends, set up a rational equation and find out how many friends actually went to the vacation home.

Solve the following radical equation involving two square roots. You will need to square both sides 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 logarithmic properties (sum of logarithms) and check the result to rule out solutions that produce negative arguments:

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