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 $24$ meters. 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.

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 meters 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 $\$100$ for shipping a package, plus an additional cost for each kilometer of distance. If for a delivery to a city located $120 \text{ km}$ away they charged $\$460$:

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 center offers a monthly membership. If 20 more people enroll than expected, the cost per person would decrease by $\$80$ to maintain the same total revenue of $\$48\,000$. 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.

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 center of Mexico City. The central space intended for lawn has dimensions of $15$ meters long by $10$ meters 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 pesos) 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 $20$ pesos, how many cases must he produce and sell per day for his net profit to be exactly $25$ pesos?

(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 $x_1 \times x_2 = c/a$).

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.