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

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

$x^2 - x - 6 = 0$

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.

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:

$2x^2 = 50$

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

To solve the equation $(2x - 3)^2 - 49 = 0$, two 5th-year students propose different approaches:

• Cian 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.

• Aoife 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 Cian's method and then solve it again following Aoife'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.

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

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.

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.

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 $x_1 \times x_2 = \frac{c}{a}$) 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.

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.

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

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.

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

Seán's age is three times that of his younger sister, Aoife. 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.

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.

A courier company charges a fixed fee of $\text{€}\ 5$ 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 $\text{€}\ 23$:

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 $350 \text{ km}$.

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

A rectangular urban garden is to be designed in the centre of Dublin. 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.

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).

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.