Exercises: Linear and quadratic equations

A content creator on a popular social network observes that the number of new followers gained each day follows a linear trend. If the number of followers $x$ satisfies the following equation, how many followers has he gained today?

$3(x - 15) + 2x = 2(x + 30)$

To complete a level in a strategy video game, you need to distribute your gold resources. The amount of gold coins $x$ that you must assign to each mine is given by solving the following equation with denominators:

$\dfrac{x}{4} - \dfrac{x - 2}{3} = -1$

Solve the following incomplete quadratic equation. Remember that, in this type of equation, factorising out a common factor is usually the fastest method to find the two solutions:

$3x^2 - 15x = 0$

A photography studio wants to print square posters for an exhibition. If twice the area of one of these posters is equal to $50$ $dm^2$, calculate the length of the side $x$ of the poster by solving the equation:

$2x^2 = 50$

Find the real solutions to the following complete quadratic equation using the quadratic formula $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$:

$x^2 - x - 6 = 0$

Find a quadratic equation of the form $ax^2 + bx + c = 0$ whose solutions are $x_1 = 3$ and $x_2 = -4$.

Javier's age is three times that of his younger sister, Elena. Knowing that the sum of their ages is $20$ years, set up a linear equation and solve it to find the age of each person.

The perimeter of a rectangular plot in a community garden measures $24$ metres. If we know that the base is twice the height, calculate the dimensions of the plot (base and height) by setting up a linear equation.

Lucía is solving the incomplete quadratic equation $3x^2 - 12x = 0$. In her notebook, she carries out the following steps: she rearranges terms to write $3x^2 = 12x$, divides both sides by $x$ obtaining $3x = 12$, and concludes that the only solution is $x = 4$.

1. a) Explain with reasons what algebraic error Lucía made by dividing by the unknown and which value of $x$ was lost in the process.
2. b) Solve the equation using the appropriate method (factoring out the common factor and setting each factor to zero) to find the complete set of real solutions.

A technical team needs to calibrate a thermal control sensor so that the system triggers two automatic alerts exactly when the variable $x$ reaches the values $x_1 = -\frac{2}{3}$ and $x_2 = 5$.

1. a) Use Vieta's formulas ($x_1 + x_2 = -\frac{b}{a}$ and $x_1 \cdot x_2 = \frac{c}{a}$) to construct a quadratic equation of the form $ax^2 + bx + c = 0$, with integer coefficients $a, b, c \in \mathbb{Z}$ such that $a > 0$ and $\operatorname{gcd}(a, |b|, |c|) = 1$.
2. b) Justify with reasons why any other equation obtained by multiplying all coefficients by a constant $k \neq 0$ keeps exactly the same solutions.

Solve the following linear equation, paying special attention to calculating the lowest common multiple and the effect of negative signs in front of fractions:

Given the quadratic equation $x^2 - 10x + k = 0$, determine with reasons the value of the parameter $k$ so that the equation has a single real solution (repeated root). Once $k$ is found, state what that solution is.

In designing a green area, a landscape architect wants to create a rectangular flowerbed whose base is 4 metres longer than its height. If the total area of the flowerbed must be $96 \text{ m}^2$, set up a quadratic equation and calculate the exact dimensions of the base and height.

Find a quadratic equation of the form $ax^2 + bx + c = 0$ knowing that the sum of its roots is $s = 7$ and the product of its roots is $p = 10$. Check your result by solving the equation obtained using the quadratic formula.

A courier company charges a fixed fee of $5 \text{ €}$ for shipping a package, plus an additional cost for each kilometre of distance. If for a delivery to a city located $120 \text{ km}$ away they charged $23 \text{ €}$:

1. Determine the linear equation relating the total cost ($C$) to the distance ($d$).

2. Calculate how much it will cost to send a package over a distance of $350 \text{ km}$.

Solve the following incomplete quadratic equations without using the quadratic formula, by factorising out a common factor or by direct rearrangement:

1. $3x^2 - 27 = 0$

2. $5x^2 + 15x = 0$

A sports centre offers a monthly membership. If 20 more people enrol than expected, the cost per person would decrease by $4 \text{ €}$ to maintain the same total revenue of $2400 \text{ €}$. Set up the corresponding equation to find out how many people were initially expected.

Determine the points of intersection between the line $y = 2x + 1$ and the parabola $y = x^2 - 2$. To do so, set both expressions equal and solve the resulting quadratic equation. Then, sketch the situation on a set of coordinate axes and mark on your drawing the intersection points you have obtained.

To solve the equation $(2x - 3)^2 - 49 = 0$, two 1st-year Bachillerato students propose different approaches:

• Marcos proposes expanding the squared binomial, subtracting $49$, combining like terms to obtain the general form $ax^2 + bx + c = 0$, and applying the quadratic formula.

• Sara proposes interpreting the expression as a difference of squares of the form $A^2 - B^2 = (A - B)(A + B) = 0$, or directly isolating $(2x - 3)^2 = 49$ and taking the square root of both sides with the double sign $\pm$.

1. a) Solve the equation following Marcos's method and then solve it again following Sara's proposal.
2. b) Critically compare both procedures: evaluate which one is more efficient in terms of number of operations and argue in which of them there is a higher probability of making calculation or sign errors.

A youth association in Valencia designs and sells T-shirts to fund a solidarity project. The fixed costs for screen printing and screens amount to $200\text{ \euro}$ and the variable manufacturing cost is $3\text{ \euro}$ per T-shirt. To boost demand, they set a unit selling price that decreases with the number of units sold $x$, according to the function $p(x) = 8 - 0,02x$ (in euros per T-shirt). The profit is zero when the total revenue $I(x) = x(8 - 0,02x)$ equals the total costs $C(x) = 200 + 3x$.

1. a) Before operating formally, make a reasoned estimate: if only $20$ T-shirts are produced, calculate mentally the total cost and compare it with the maximum possible revenue ($8\text{ \euro}$ per unit). Will there be a loss or a profit? Estimate in what range of produced T-shirts the initiative will start to be profitable.
2. b) Set up the equation $I(x) = C(x)$, reduce it to a quadratic equation, and solve it to find the exact break-even points (zero profit).
3. c) Check whether the two mathematical solutions obtained are consistent with your previous estimate and explain why, in this economic model, producing more than $200$ T-shirts generates losses again.

In a quadratic equation of the form $x^2 - (k+3)x + 3k = 0$, determine the values that the real parameter $k$ must take so that the two roots of the equation are consecutive integers. Once $k$ is found, indicate what these solutions would be for each value found.

A digital marketing agency has shown that the success of a social media campaign (measured in thousands of "likes") follows a rational model depending on time $x$ (in weeks) according to the following expression:

Find the value of $x$ that satisfies this equation. Do not forget to check whether the values obtained are valid within the domain of the given rational expressions.

A rectangular urban garden is to be designed in the centre of Madrid. The central space intended for lawn has dimensions of $15$ metres long by $10$ metres wide. Around this space, a gravel path of uniform width $x$ is to be built. Calculate the value of this width $x$, knowing that the total area of the gravel path must be exactly equal to three-quarters of the area of the lawn alone.

Solve the following equation with nested radicals and make sure to verify the validity of the solutions obtained in the original expression:

A young entrepreneur manufactures custom phone cases using a 3D printer. The total daily cost $C$ (in euros) is given by the expression $C = x^2 - 10x + 200$, where $x$ is the number of cases produced. If he sells each case at a fixed price of $20$ euros, how many cases must he produce and sell per day for his net profit to be exactly $25$ euros?

(Remember that Profit = Total Revenue - Total Costs).

Given the quadratic equation $x^2 - mx + 36 = 0$, where $m$ is a real number, determine the value of this coefficient $m$ knowing that the sum of the reciprocals of its roots, $x_1$ and $x_2$, is equal to $\frac{5}{12}$.

(Hint: Use Cardano's relations between the coefficients and the roots: $x_1 + x_2 = -b/a$ and $x_1 \cdot x_2 = c/a$).

Find the solutions to the following rational equation, paying special attention to the simplification of terms and the possible appearance of extraneous solutions:

A model aircraft club organises a flight test: a drone must make a round trip between two control points located at a distance of $12$ km. The drone flies at a constant airspeed. On the outward journey, the drone has a tailwind of $10$ km/h, but on the return journey it faces the same headwind. Knowing that the drone took a total of $54$ minutes to complete the round trip, calculate the drone's airspeed (in km/h) in still air.

According to Adif records, the high-speed rail line between Madrid and Valencia is $390\text{ km}$ long. Under normal operating conditions, an AVE train covers this route at a constant average speed $v$ (expressed in $\text{km/h}$).

Due to a warning for strong wind gusts issued by AEMET, on a specific day the train must preventively reduce its average speed by $65\text{ km/h}$ throughout the entire journey, which causes a delay in arrival at the destination of exactly $30\text{ minutes}$ relative to the official timetable.

1. a) Express in hours the usual travel time $t_1$ and the travel time at reduced speed $t_2$ in terms of the unknown $v$.
2. b) Write the equation that models the delay ($t_2 - t_1 = 0,5\text{ h}$), clear the denominators, and obtain the quadratic equation in its standard form $av^2 + bv + c = 0$.
3. c) Solve the equation, calculate the train's usual average speed along with the scheduled travel time, and justify why one of the mathematical solutions obtained must be discarded.

In the risk department of a financial institution, the stability of two investment portfolios is being analyzed, whose critical rates of return $x$ satisfy an equation that depends on a real parameter $k$.

A junior analyst states in their report: "In the equation $x^2 - 2kx + (k^2 + 9) = 0$, it is sufficient to choose a sufficiently large positive value of $k$ to ensure that there are two distinct real returns, since as $k$ increases, the linear term overcomes the constants."

1. a) Critically analyze the analyst's statement: calculate the discriminant $\Delta$ in terms of $k$ and prove with reasons why their conclusion is false for any value of $k \in \mathbb{R}$.
2. b) The team corrects the model and proposes the equation $x^2 - 2kx + (k^2 - 9) = 0$. Use the discriminant to prove that this new equation indeed has two distinct real solutions for any $k \in \mathbb{R}$.
3. c) Without explicitly calculating the solutions $x_1$ and $x_2$ using the quadratic formula, apply Vieta's formulas regarding the sum ($x_1 + x_2$) and the product ($x_1 \cdot x_2$) to determine with justification the interval of values of the parameter $k$ that ensures both rates of return are strictly positive ($x_1 > 0$ and $x_2 > 0$).