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:

$\log_2(x + 5) = 3$

Daniel has to solve the radical equation $\sqrt{2x + 5} = x - 5$ in class. To eliminate the square root, he squares both sides and obtains $2x + 5 = x^2 - 10x + 25$. After simplifying and moving all terms to one side, he arrives at $x^2 - 12x + 20 = 0$. He applies the quadratic formula and finds two solutions: $x = 10$ and $x = 2$. Very confident, he writes on his exam that both are solutions to the initial equation.

1. a) Explain what logical property fails when squaring both sides of an equality and why this step can introduce "extraneous solutions".
2. b) Without solving the quadratic equation again, numerically check the two values in the original equation $\sqrt{2x + 5} = x - 5$ and state which is the only valid solution, explaining why the other must be discarded.

We want to construct a biquadratic equation of the form $x^4 + bx^2 + c = 0$ that simultaneously satisfies two conditions:

1. Its only real solutions must be $x = 3$ and $x = -3$.

2. When applying the change of variable $t = x^2$, the resulting quadratic equation in the variable $t$ must have two distinct real solutions: $t_1$ and $t_2$.

1. a) Reason what sign and what exact numerical value one of the solutions for $t$ must have, and what sign the other solution must necessarily have so that it does not provide any more real roots for the unknown $x$.
2. b) Choose a numerical value for that second solution for $t$ and write the complete resulting biquadratic equation $x^4 + bx^2 + c = 0$, verifying that it satisfies the required conditions.

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 $v_m = \frac{2 \cdot 20 \cdot x}{20 + x}$. 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) that a car travels 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, at what speed was it travelling?

The number of members on a Discord server follows the exponential formula $N = 500 \cdot 2^{t/3}$, 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:

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

Consider the rational equation $\frac{12}{x} - \frac{12}{x + 2} = 1$.

1. a) Before operating, make a reasoned estimation to look for a positive solution $x > 0$:

• If $x = 2$, mentally calculate the value of $\frac{12}{x} - \frac{12}{x + 2}$ and compare it with $1$.

• If $x = 6$, mentally calculate the value of that same expression and compare it with $1$.

• From these two tests, deduce and justify between which two integers the positive solution must lie.

1. b) Formally solve the equation by clearing denominators using the least common multiple, check whether your estimation is confirmed, and determine all real solutions of the equation, pointing out which values of $x$ were forbidden from the beginning.

To solve exponential equations, two main strategies are usually used: factoring out a common factor or performing a change of variable of the type $u = a^x$. Look at these two equations:

(1) $2^{x + 3} - 2^x = 56$

(2) $4^x - 6 \cdot 2^x + 8 = 0$

1. a) Solve equation (1) by factoring out the power $2^x$ on the left-hand side and justify why this path is more direct than making a change of variable.
2. b) Explain why the method of factoring out a common factor is not directly applicable to equation (2) (paying attention to the presence of the constant term and the bases of the powers).
3. c) Solve equation (2) using the change of variable $u = 2^x$, clearly indicating the final solutions for the unknown $x$.

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 euros) 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 which times $t$ is the value of the investment exactly $2$ thousand euros? (Hint: use the laws of indices 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 $A_2 = 100 \cdot A_1$, 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 holiday home for the weekend for a total of $120$ €. At the last minute, $2$ more friends join the trip, which means each person who was originally going pays $3$ € less. If $x$ is the initial number of friends, set up a rational equation and find out how many friends actually went to the holiday 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:

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

A Renfe medium-distance train makes a round trip between two cities separated by $60\text{ km}$. On the outbound trip, due to a temporary speed restriction caused by track maintenance work, the train travels at an average speed of $v - 20\text{ km/h}$, where $v$ is its usual average speed. On the return trip, without any restrictions, it travels at its usual speed of $v\text{ km/h}$. It is known that the outbound trip took exactly 15 minutes ($0,25\text{ hours}$) longer than the return trip.

1. a) Set up a rational equation that models the situation based on the relationship between time, distance, and speed, identifying the meaning of each term and the restrictions that physical reality imposes on the value of $v$.
2. b) Solve the equation to calculate the value of the usual speed $v$ and the actual speed at which it traveled on the outbound trip.
3. c) Calculate the total time the train was in motion throughout the entire journey (outbound and return), expressing the result in hours and minutes.

We wish to design a logarithmic equation of the form $\log(x + a) + \log(x + 1) = \log(b)$, where $a$ and $b$ are fixed real constants, so that the following occurs:

• When applying the properties of logarithms and transforming the equality into a quadratic polynomial equation, the algebraic solutions to that equation are exactly $x = 5$ and $x = -3$.

• However, when checking in the original logarithmic equation, the value $x = -3$ must be discarded (because it does not belong to the domain of definition), so that $x = 5$ is the only valid solution.

1. a) From the relationships between the roots and coefficients of a quadratic polynomial, deduce the required numerical values of the constants $a$ and $b$.
2. b) Explain precisely the apparent paradox: why does the value $x = -3$ manage to satisfy the product polynomial equation but turns out to be inadmissible in the initial logarithmic equation?
3. c) Verify that, with the obtained values of $a$ and $b$, the value $x = 5$ fully satisfies the initial logarithmic equation.