Exercises: Other types of equations

Solve the following biquadratic equation, finding all its real solutions:

$x^4 - 13x^2 + 36 = 0$

Solve the following rational equation, paying attention to possible extraneous solutions that make the denominators zero:

$\frac{x+2}{x-1} = \frac{x}{x+3}$

Find the value or values of $x$ that satisfy the following radical equation. Remember to check the validity of the solutions obtained:

$\sqrt{2x+5} - 3 = x$

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

$3^{2x-1} = 81$

Find the value of $x$ by applying the laws of logarithms in the following equation:

$\log(x+2) + \log(3) = \log(15)$

Find all real solutions to the following polynomial equation, which is already factorised:

$(x^2-9)(2x+5)(x^2+4) = 0$

Solve the following incomplete biquadratic equation by taking out a common factor:

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

Isolate the radical, square both sides and solve the irrational equation. Check whether the solutions found are valid:

$\sqrt{x^2+5x+1} = x+2$

Lucía has tried to solve the following rational equation on the board:

To eliminate the denominators, she multiplied all terms by the lowest common multiple, which is $(x-2)$, working through the following steps:

Upon finishing, she firmly states that the unique solution to the equation is $x = 2$.

1. a) Without re-doing the algebraic operations, substitute $x = 2$ into the original equation and explain what happens to the denominators.
2. b) Explain what Lucía's conceptual error was in accepting that solution as valid and determine what the solution set of this equation actually is.

In a central post office in Zaragoza, they have two parcel sorting machines: a new high-speed model and an old model. The new machine can sort all the shift's mail in $3\text{ hours}$ less than the old machine takes working alone. If the operators run both machines simultaneously, they manage to finish the entire job in exactly $2\text{ hours}$.

1. a) If we call $t$ the time (in hours) that the old machine takes to sort the entire shift alone, write the fraction of the total work done by each machine in a single hour.
2. b) Set up a rational equation representing the combined work in one hour and solve it to calculate how many hours each of the two machines would take if working separately (remember to justify discarding any solutions that lack physical meaning).

The area of a rectangle is $120\text{ cm}^2$ and the length of its diagonal is $17\text{ cm}$. If the base is $x$ and the height is $y$, set up a system of non-linear equations. Reduce the system to a biquadratic equation with a single unknown and find the dimensions of the rectangle.

A cyclist travels a distance of $60\text{ km}$ at a constant speed $v$. If they had ridden $5\text{ km/h}$ faster, it would have taken them $1$ hour less to complete the same journey. Knowing that the time taken is $t = \frac{d}{v}$, set up a rational equation and determine the initial speed of the cyclist.

The number of views of a video that is going viral grows according to the function $N(t) = 100 \cdot 2^{t/3}$, where $N$ is the cumulative number of views and $t$ is the time elapsed in hours since its publication. How many hours must pass for the video to reach exactly $6400$ views?

The sound intensity level $\beta$ in decibels (dB) is calculated using the expression $\beta = 10 \log\left(\frac{I}{I_0}\right)$, where $I$ is the sound intensity of the noise and $I_0 = 10^{-12}\text{ W/m}^2$ is the threshold of hearing. If the traffic noise in a busy street reaches $80\text{ dB}$, find the sound intensity $I$ by solving the corresponding logarithmic equation.

The period of oscillation $T$ of a simple pendulum, measured in seconds, is given by the formula $T = 2\pi \sqrt{\frac{L}{g}}$, where $L$ is the length of the string in metres and $g = 9,8\text{ m/s}^2$ is the acceleration due to gravity. If a pendulum clock has an exact period of $2\text{ s}$, set up and solve the irrational equation to calculate the length $L$ of the pendulum. (Use the approximation $\pi \approx 3,14$).

Solve the following exponential equation by applying the substitution $u = 2^x$. First obtain the values of $u$ and then reverse the substitution to find the values of $x$:

$4^x - 5 \cdot 2^x + 4 = 0$

An open-topped box is to be built from a square piece of cardboard with side length $20\text{ cm}$. To do this, squares of side length $x$ are cut out from each of the four corners, and the flaps are folded upwards. The volume of the resulting box is given by the expression $V(x) = x(20-2x)^2$. Set up the necessary polynomial equation and find the integer value of $x$ such that the volume of the box is $576\text{ cm}^3$.

Solve the following irrational equation involving two radicals. Isolate one of them, square both sides, and repeat the process if necessary. Do not forget to check the validity of your results in the original equation:

$\sqrt{x+4} + \sqrt{x-1} = 5$

We want to construct a biquadratic equation of the form $a\cdot x^4 + b\cdot x^2 + c = 0$ for a Mathematics exam, such that it satisfies all the following conditions:

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

2. It must have no other real solutions (therefore, when applying the substitution $z = x^2$, one of the solutions for $z$ must generate the real solutions and the other must be strictly negative so as not to provide real roots).

3. The leading coefficient must be $a = 1$ and the constant term must be $c = -36$.

1. a) Deduce, with reasoning, what value the second root of the auxiliary variable $z$ must have, and determine the numerical value that coefficient $b$ must take.
2. b) Write the resulting complete biquadratic equation and solve it step by step to verify that it indeed only admits $x = 3$ and $x = -3$ as real solutions.

The Traffic Division of the Guardia Civil and the DGT use mathematical models to estimate the speed at which a car was travelling before emergency braking based on the tire skid mark left on dry asphalt. On a specific stretch of a national road, the relationship between speed $v$ (in $\text{km/h}$) and the length $d$ of the skid mark (in metres) is given by the radical equation:

In an accident report following a speeding incident, the speed camera recorded that the car was travelling at exactly $v = 60\text{ km/h}$.

1. a) Estimate: Before making precise measurements, a witness states that the skid mark measured at least $50\text{ metres}$. Substitute $d = 50$ mentally or with a quick calculation into the formula to estimate what speed would correspond to that length, and reason whether the witness's claim is credible or excessive.
2. b) Calculate and check: Set up the equation by substituting $v = 60\text{ km/h}$, solve for the unknown $d$ by isolating the square root and squaring both sides, and verify that the solution obtained is not an extraneous solution in the original equation. How many metres did the skid mark actually measure?

Solve the following modulus equation, analysing the necessary cases depending on the sign of the argument and checking the validity of the solutions obtained:

$|2x - 5| = x + 1$

When solving the exponential equation $3^{x+1} - 3^{x-1} = 24$, a student tries to take out $3^x$ as a common factor but makes an error in factorising, writing the expression as $3^x \cdot (3^1 - 3^0) = 24$, arguing that "when taking $x$ out of the exponent $x-1$, $0$ remains". Identify the error in applying the laws of indices with negative exponents, correct the extracted common factor, and solve the equation to find the exact value of $x$.

Given the biquadratic equation $x^4 - 8x^2 + k = 0$, where $k$ is a real parameter:

Set up the substitution $z = x^2$ and analyse the discriminant of the resulting quadratic equation. Determine what condition or values parameter $k$ must satisfy for the original equation to have exactly four distinct real solutions.

Find the points of intersection between a circle centred at the origin with radius $5$, whose equation is $x^2+y^2 = 25$, and the inverse proportion curve $x \cdot y = 12$. Set up the system of non-linear equations, reduce it to a biquadratic equation, and find the coordinates $(x,y)$ of all intersection points.

Two taps, A and B, fill a water tank working together in exactly $6$ hours. It is known that, working separately, tap A would take $5$ hours less than tap B to completely fill the same tank. Set up a rational equation relating the flow rates (fraction of tank per hour) and find how long each tap would take individually.

An investor deposits a capital of $5000$ euros into a financial product at an annual compound interest rate of $r\%$. The formula for the final capital after $t$ years is $C_f = C_i \left(1 + \frac{r}{100}\right)^t$. After $2$ years, the capital has grown to $5408$ euros. Set up the corresponding exponential or polynomial equation and determine the annual interest rate $r$.

In a right circular cone, the slant height $g$, the base radius $r$, and the height $h$ form a right-angled triangle, satisfying $g = \sqrt{r^2+h^2}$. It is known that the base radius is $3\text{ cm}$ less than the height of the cone, and the slant height is exactly $15\text{ cm}$. Set up the corresponding irrational equation and solve it to find the exact measurements of the radius and the height.

Solve the following system of logarithmic equations by applying the laws of logarithms to transform it into a linear or non-linear algebraic system:

$\begin{cases} \log(x) + \log(y) = 3 \\ 2\log(x) - \log(y) = 3 \end{cases}$

Solve the following system of exponential equations. Use a suitable substitution, for example $u = 2^x$ and $v = 5^y$, to transform it into a linear system, solve it, and then reverse the substitution to find $x$ and $y$:

$\begin{cases} 2^x + 5^y = 9 \\ 2^{x+2} - 5^{y+1} = -9 \end{cases}$

Find all the roots of the following fourth-degree polynomial by solving the corresponding equation. Use the Remainder Theorem and Ruffini's rule to find the integer roots, and then solve the resulting quadratic equation to find any remaining real solutions or check if there are any:

$x^4 - 2x^3 - 7x^2 + 8x + 12 = 0$

The class is asked to solve the radical equation with two radicals:

Two classmates propose different strategies for their first step:

• Marta's strategy: Isolate one radical by moving the other to the right-hand side by addition, $\sqrt{x+7} = 1 + \sqrt{x}$, and then square both sides.

• Guille's strategy: Square the equation directly as written, $(\sqrt{x+7} - \sqrt{x})^2 = 1^2$, applying the algebraic identity for the square of a difference.

1. a) Carry out the algebraic expansion of that first step using both strategies, simplifying like terms.
2. b) Compare the two results obtained: explain in detail which of the two strategies is more advantageous and which generates an expression that is more complex or carries a higher chance of sign errors when isolating the remaining radical.
3. c) Finish solving the equation using the most efficient method and check whether the value obtained is indeed a solution to the original equation, justifying why this checking step is mandatory when solving radical equations.

A $100\,\%$ electric delivery van travels the intercity route between Logroño and Vitoria, covering a distance of $60\text{ km}$ each way ($120\text{ km}$ in total for the round trip).

• On the outbound journey, due to smooth morning traffic, it travels at an average speed $v$ (in $\text{km/h}$).

• On the return journey, with heavier congestion and traffic jams at the entrance, its average speed is reduced by $20\text{ km/h}$.

The driver records that the total time taken to complete both journeys (outbound and return) was exactly $2\text{ hours and } 30\text{ minutes}$ ($2,5\text{ hours}$).

1. a) Express the outbound time and the return time in terms of $v$, and set up the rational equation that equates the sum of both times to the total journey time.
2. b) Solve the resulting rational equation, analyse the two possible mathematical solutions, and interpret which one makes no sense in the context of the problem, stating the outbound and return speeds.
3. c) To optimize battery consumption, the logistics company requires that the overall average speed of the entire trip (defined as the total distance covered divided by the total time taken) be at least $45\text{ km/h}$. Prove with reasoning whether or not this requirement is met, noting why calculating the overall average speed as the simple arithmetic mean of the outbound and return speeds would be a common calculation error.