Other equations

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

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

Design a biquadratic equation of the form $x^4 + bx^2 + c = 0$ (with integer coefficients $b, c \in \mathbb{Z}$) that simultaneously satisfies two conditions:

1. Its only real solutions must be $x = 2$ and $x = -2$.

2. The auxiliary quadratic equation in $t$ ($t = x^2$) must have two distinct real solutions (that is, strictly positive discriminant, $\Delta > 0$).

Show step by step how you deduce the values of $b$ and $c$, justifying what sign each root of the auxiliary variable $t$ must have and why that choice guarantees that the original equation does not have four real solutions.

The diagonal of a rectangle measures $10$ cm and its area is $48\text{ cm}^2$. Set up a biquadratic equation to find the lengths of its sides and solve it.

Determine the value of $x$ that satisfies the following exponential equation:

$3^{2x+1} - 27 = 0$

A bacterial culture grows according to an exponential law. To determine the critical moment of an experiment, a biologist must solve the equation $4^x - 5 \times 2^x + 4 = 0$. Find the values of $x$ by applying a substitution.

Consider the exponential equation $4^x + 2^{x+1} = 80$.

1. a) Before solving it algebraically, find a reasoned bound showing between which two integers the real solution lies. To do so, mentally evaluate the left-hand side for $x = 2$ and for $x = 4$, and justify this bound based on the strictly increasing nature of exponential functions.
2. b) Formally solve the equation using the substitution $u = 2^x$. Precisely justify which root of the auxiliary quadratic equation is discarded and on what theoretical grounds.
3. c) Check whether the exact value obtained matches your previous estimate and verify the numerical equality in the original equation.

Apply the properties of logarithms to solve the equation:

$\log_2(x+3) + \log_2(x-3) = 4$

Seán has to solve the logarithmic equation $\log(x) + \log(x - 3) = 1$ (where the logarithm is in base 10) in an exam. On his paper, he writes the following steps:

1. I apply the product rule for logarithms: $\log[x(x - 3)] = 1$.

2. By the definition of logarithm: $x(x - 3) = 10^1 \implies x^2 - 3x - 10 = 0$.

3. I solve the quadratic equation: $(x - 5)(x + 2) = 0 \implies x_1 = 5$ and $x_2 = -2$.

4. Check: substituting $x = -2$, we have $(-2)\times(-2 - 3) = (-2)\times(-5) = 10 > 0$; since the resulting argument is positive, both solutions are valid.

1. a) Identify the conceptual error made by Seán and explain why his argument in step 4 is incorrect when applied to the original equation.
2. b) Determine the domain of validity of the original equation and write the final set of valid real solutions.

Apply the properties of logarithms to group the terms and solve the equation:

$\log(x+1) - \log(x-1) = \log(x+4) - \log(x+1)$

Factorise the polynomial and solve the cubic equation:

$x^3 - 4x^2 - x + 4 = 0$

An open-topped box is to be constructed from a rectangular piece of cardboard measuring $20 \times 15$ cm, by cutting out equal squares of side length $x$ from the corners and folding up the sides. If the volume of the box must be $378\text{ cm}^3$, set up the corresponding cubic polynomial equation and find the value of $x$, knowing that it is an integer.

To solve the polynomial equation $x^3 - 3x^2 - 4x + 12 = 0$, two classmates propose different approaches:

• Aoife proposes looking for an integer root by testing among the divisors of 12 and factoring using Ruffini's rule.

• Seán suggests grouping terms in pairs: $(x^3 - 3x^2) - (4x - 12) = 0$, and factoring out consecutive common factors without the need to test divisors.

1. a) Solve the equation following Aoife's method, showing the Ruffini table and the resulting quadratic equation.
2. b) Solve the equation following Seán's method, detailing each factoring out of a common factor.
3. c) Compare both procedures: argue which one is more efficient in this specific case and explain whether Seán's method would be applicable to any cubic equation or if it requires a particular relationship between the coefficients (support your answer with a counterexample).

A card's PIN consists of four positive, consecutive integers in increasing order. If we construct a monic polynomial $P(x)$ of degree $4$ whose roots are precisely those four digits, its constant term turns out to be equal to $24$. Determine the expanded polynomial $P(x)$ and the four-digit PIN.

Find the value of $x$ that satisfies the following equation with the unknown in the denominator:

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

Find the real solutions to the irrational equation, checking their validity:

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

Find the values of $x$ that satisfy the equation with absolute value:

$|2x - 5| = 7$

Solve the following rational equation by reducing it to a polynomial equation:

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

Two hoses fill a tank in $6$ hours working together. If the first hose takes $5$ hours less than the second to fill the tank on its own, how long does each one take? Set up and solve the corresponding rational equation.

Solve the following irrational equation with two radicals and check the solution obtained:

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

Find all real values of $x$ satisfying the following equation with quadratic modulus:

$|x^2 - 5x + 4| = 2$

Determine for which values of the parameter $k$ the biquadratic equation $x^4 - kx^2 + 9 = 0$ has exactly two distinct real solutions. Give full justification for your answer.

A cyclist travels from town A to town B, separated by $60$ km, at a constant speed. On the return journey, due to a headwind, their speed decreases by $5$ km/h, taking $1$ hour longer to complete the same journey. Set up the rational equation that models the situation and determine the speed on the outward journey.

A telecommunications tower is located on the straight bank of a river. A cable is to be stretched in a straight line from a point $P$ on the bank to the tower, and another cable from $P$ to a cabin located $5$ km downstream from the tower and $1$ km inland. If the total length of cable used is $6$ km, set up a radical equation and determine the distance from point $P$ to the tower.

Find all real solutions of the equation with nested modulus:

$||x - 2| - 3| = 1$

Consider the family of exponential equations depending on the real parameter $k$:

1. a) Apply the substitution $u = 2^x$ to transform the equation into a quadratic one, and explicitly state the algebraic restriction that any solution for $u$ must satisfy to generate a real solution for $x$.
2. b) Discuss reasonably the number of real solutions of the original equation depending on the values of the parameter $k$. Determine precisely for which values or intervals of $k$ the equation:

1. Has exactly two distinct real solutions.

2. Has a unique real solution.

3. Has no real solution.

(Justify your classification by analyzing the discriminant of the equation in $u$ and the signs of its roots using Cardano-Vieta's sum and product).

1. c) For the particular case $k = 3$, calculate all the exact solutions of the equation (using logarithms when it is not an integer power of the base).

Topic Challenge. How does the weather influence your home's energy consumption? Design a mathematical model to optimize the use of air conditioning according to outdoor temperature.

Phase 1. Real data. Look up on the Met Éireann website the maximum and minimum temperatures in your municipality over the last three days. Calculate the average ($T_m$) and set up an absolute value equation, $|T - T_m| = k$ with $k = 3$ °C, to find the two critical thermal comfort thresholds.

Phase 2. Algebraic solution.

1) Model the average hourly cost based on hours of use $t$ using a rational equation: $\frac{12t + 35}{t} = 14.5$. Solve for $t$ and analyze its feasibility.

2) Model heat flux with an irrational (radical) equation that leads to a quadratic one: $\sqrt{2x + 7} - x = 2$. Solve analytically in your notebook, verifying and discarding potential extraneous solutions.

Phase 3. GeoGebra and conclusions. Plot the associated functions in GeoGebra (absolute value, rational, and the parabola derived from the radical) to visualize the points of intersection. Write a short report with your graphs and a reasoned recommendation for household consumption.

Assessment criteria:

• Precision in the analytical solution of the equations and discrimination of extraneous solutions.

• Correct use of GeoGebra to geometrically contrast the algebraic results.

• Consistency and rigour of the report when linking Met Éireann data to the final proposal.

A sample of radioactive material decays such that the remaining amount $M(t)$ in grams after $t$ years obeys the expression $M(t) = 200 \times e^{-0.05t}$. How many years must pass for exactly $50$ grams of material to remain? Express the exact result using natural logarithms.

The energy $E$ (in ergs) released by an earthquake is related to its magnitude $M$ on the Richter scale by the equation $\log E = 11.8 + 1.5M$. If an earthquake releases $1000$ times more energy than another whose magnitude is $4.2$, what is the magnitude of the first earthquake on the Richter scale?

Solve the following non-linear system modelling the intersection of two paths in the Cartesian plane: $y = \sqrt{x+2}$ and $y = |x|$. Determine the points of intersection algebraically.

On a farm in the Golden Vale region (County Tipperary), an open-air rectangular irrigation pond is planned to optimize water storage during drought periods.

The technical design establishes that the length of the base will exceed its width $x$ by $8\text{ meters}$. For geotechnical reasons related to slope stability, the depth (height) of the pond must be exactly equal to half its width.

1. a) Knowing that the required net volume is $96\text{ m}^3$, set up the polynomial equation in its general form $ax^3 + bx^2 + cx + d = 0$ that models the capacity of the pond as a function of its width $x$.
2. b) Solve the equation by factoring the polynomial using the remainder theorem and Ruffini's rule. Prove, by analyzing the discriminant of the quotient polynomial, that no other real dimensions are possible.
3. c) State the final dimensions of the pond (width, length, and depth).
4. d) If a subsequent technical regulation limited the maximum depth to $1.5\text{ meters}$ while maintaining the same design relationships (length $x+8$ and depth equal to half the width), argue algebraically (without needing to solve a new complete equation) why it would be impossible to reach the required volume of $96\text{ m}^3$.