Exercises: Quadratic equation word problems

Find a positive number such that three times its square is exactly equal to twelve times the number itself.

The product of two consecutive integers is $156$. What are these numbers?

The base of a rectangular plot measures $5$ metres more than its height. If its total area is $150 \text{ m}^2$, calculate its dimensions.

If we subtract three times Lucía's age $2$ years ago from the square of the age she will be in $3$ years, we get $55$. How old is Lucía now?

In a right-angled triangle, one leg measures $7 \text{ cm}$ more than the other. If the hypotenuse measures $13 \text{ cm}$, calculate the length of both legs.

The product of two consecutive positive even numbers is $288$. Find these numbers.

The sum of the squares of two consecutive odd natural numbers is $130$. Determine which numbers these are.

If the side of a ceramic square tile is increased by $4 \text{ cm}$, its area quadruples. How long was the side initially?

Marcos solves the following problem on his exam: “The floor of a rectangular warehouse has a length that measures $5\text{ m}$ more than its width. If the total area is $66\text{ m}^2$, what are its dimensions?”.

To solve it, he sets up $x(x+5) = 66 \implies x^2 + 5x - 66 = 0$. He applies the quadratic formula and correctly obtains the algebraic roots $x_1 = 6$ and $x_2 = -11$. Then, he concludes: “There are two possible warehouses: one $6\text{ m}$ wide by $11\text{ m}$ long, and another $11\text{ m}$ wide by $16\text{ m}$ long, because since distances cannot be negative, we change the sign of $-11$ and add $5$ to it”.

1. a) Precisely identify Marcos's conceptual error when interpreting the negative solution $x_2 = -11$.
2. b) Explain why the algebraic equation yields two solutions while the geometric problem only admits one.
3. c) Correct the final answer by determining the true dimensions of the warehouse and calculate how many linear metres of skirting board will be needed to edge its entire perimeter (without discounting doors).

A ceramic artisan from Talavera de la Reina designs a square decorative tile. For a special order, he modifies the mould by increasing its base by $3\text{ cm}$ and reducing its height by $2\text{ cm}$, obtaining a rectangular piece whose area is $84\text{ cm}^2$.

1. a) Without solving the equation, make a preliminary estimate: taking into account that the final area ($84\text{ cm}^2$) must be very similar to that of the initial square, estimate between which two consecutive integers the length of the tile's original side must lie.
2. b) Define the unknown, set up the corresponding quadratic equation, and solve it step by step to find the exact measurement of the side of the original tile.
3. c) Check whether your mathematical solution matches your initial estimate and verify against the problem statement that the dimensions of the new rectangle satisfy the specified area.

A rectangular swimming pool measuring $10 \text{ m}$ long by $6 \text{ m}$ wide is surrounded by a grass border of uniform width $x$. If the area of the border alone is $80 \text{ m}^2$, what is the width of the border?

To make a kite, a piece of fabric in the shape of a rhombus is needed. It is known that the major diagonal measures $4 \text{ cm}$ more than the minor diagonal, and that the total area of the kite is $96 \text{ cm}^2$. Calculate the length of both diagonals.

A group of friends decides to buy a wedding present costing $120$ euros, sharing the cost equally. At the last minute, two more friends join the group, which means each of the previous members pays $3$ euros less than initially planned. How many friends were in the initial group?

A projectile is launched vertically upwards. Its height $h$ (in metres) as a function of time $t$ (in seconds) is given by the mathematical expression $h = 40t - 5t^2$. At what times will the projectile be at a height of exactly $75 \text{ metres}$?

The cost price of a batch of T-shirts is $300$ euros. The seller decides to keep $5$ T-shirts for their family and sell the rest for $2$ euros more than each cost. If, after selling them all, they make a net profit of $60$ euros (that is, receiving a total of $360$ euros in revenue), how many T-shirts were bought initially?

A tech company manufactures rectangular solar panels. The length of the panel must be twice its width minus $1 \text{ metro}$. If a new model requires the surface area of the panel to be $28 \text{ m}^2$, find its dimensions.

The difference between the ages of a grandfather and his grandson is $56$ years. $4$ years ago, the grandfather's age was exactly the square of the grandson's age. How old are they now?

In an athletics competition, a shot put is thrown. The trajectory follows a parabola given by the equation $y = -0,05x^2 + 0,8x + 1,8$, where $x$ is the horizontal distance in metres and $y$ is the height in metres. At what horizontal distance will it hit the ground?

To promote sustainable mobility, a municipality designs a new enclosed bicycle car park with an area of $180\text{ m}^2$. Due to pedestrian pavement constraints, the depth (width) of the enclosure must measure exactly $8\text{ m}$ less than its facade (length).

1. a) Let $x$ be the length of the facade in metres, write the expression that represents the depth, and set up a single quadratic equation that models the area.
2. b) Solve the equation, analyse the meaning of the two mathematical solutions obtained, and determine the real dimensions of the plot.
3. c) The town council has a maximum budget of $8000\text{ euros}$. If the permeable paving costs $35\text{ euros/m}^2$ and the perimeter security fence enclosing all four sides costs $25\text{ euros}$ per linear metre, check numerically whether the allocated budget is sufficient to carry out the project.

A teacher wants to prepare a geometric design problem for their class whose resolution leads to the quadratic equation $x^2 - 15x + 54 = 0$, whose solutions are $x_1 = 9$ and $x_2 = 6$.

1. a) Write the statement of a contextualised problem (for example, fencing a vegetable garden or designing a terrace) involving the semiperimeter ($15\text{ m}$) and the area ($54\text{ m}^2$), so that setting it up directly yields said equation.
2. b) Solve the equation and interpret why in this particular case BOTH numerical solutions are positive and valid, explaining what they represent geometrically.
3. c) Modify only the area value in your problem statement to create a physically impossible situation (that is, where no rectangle can be constructed with that semiperimeter). Justify your proposal by analysing the sign of the discriminant ($\Delta = b^2 - 4ac$) of the resulting new equation.

A rectangular cardboard sheet measuring $30 \text{ cm}$ long by $20 \text{ cm}$ wide is available. It is desired to construct an open box by cutting a square of side $x$ from each corner and folding up the flaps. If the base area of the resulting box must be $336 \text{ cm}^2$, calculate the value of $x$ and the volume of the box.

Two pipes together fill a water tank in $6$ hours. If only the first pipe were opened, it would take $5$ hours less to fill the tank than if only the second were opened. How long would each pipe take to fill the tank independently?

Two cars leave the same junction at the same time along two perpendicular roads. One travels $10 \text{ km/h}$ faster than the other. After $2$ hours, the straight-line distance between them is $100 \text{ km}$. At what speed is each car travelling?

An investor invests a principal of $5000$ euros at a certain annual compound interest rate. After two years, the accumulated amount is $5408$ euros. What was the annual interest rate applied?

A cyclist travels from town A to town B, which are $60 \text{ km}$ apart. On the return journey, due to fatigue, their average speed is $5 \text{ km/h}$ slower than on the outward journey, so it takes $1$ hour longer to complete the same distance. At what speed did they travel on the outward journey?

An artisan wants to design a wooden frame for a rectangular mirror. The dimensions of the mirror are $80 \text{ cm}$ high by $60 \text{ cm}$ wide. The frame will have a uniform width of $x \text{ cm}$ at the top and bottom, but at the sides the width will be double, that is, $2x \text{ cm}$. If the total area of the frame (excluding the mirror) must be $2400 \text{ cm}^2$, determine the value of $x$.

A theatre company hires a hall for $800$ euros for a performance. To cover expenses, they decide to set a ticket price. On the day of the performance, $40$ more audience members attend than expected, so the organiser decides to lower the ticket price by $1$ euro, raising exactly the necessary $800$ euros. How many audience members actually attended the performance?

A cylindrical tin can is designed so that its height is exactly equal to the diameter of its base. If exactly $150\pi \text{ cm}^2$ of tinplate was used to manufacture the can (including the two circular lids and the curved surface), calculate the radius of the base.

A community of neighbours in Logroño has a square interior courtyard of unknown side length. In a renovation, they decide to reduce one side by $2\text{ m}$ to install an accessibility ramp and extend the perpendicular side by $6\text{ m}$ by taking advantage of a disused storage room. The resulting new rectangular courtyard has an area of $105\text{ m}^2$.

To find the dimensions of the original courtyard, two committee members propose different approaches:

• Method A (Carlos): Chooses the side of the original square courtyard as the unknown $x$ and sets up the equation: $(x - 2)(x + 6) = 105$.

• Method B (Lucía): Chooses the smaller width of the new rectangular courtyard as the unknown $y$, deduces that the length will measure $y + 8$, and sets up the equation: $y(y + 8) = 105$, calculating the original side at the end as $y + 2$.

1. a) Expand and solve both methods completely. Verify that both lead to the same initial and final dimensions.
2. b) Algebraically compare both equations: which one has more manageable coefficients for calculating the discriminant mentally? What are the advantages and disadvantages of each in terms of the risk of oversight when answering the problem's final question?

A high-speed train regularly covers the direct $300\text{ km}$ journey between Madrid and Zaragoza at a constant average speed $v$ (in $\text{km/h}$). Due to track work, on a certain day it must reduce its average speed over the entire route by $50\text{ km/h}$, causing a delay of half an hour ($0,5\text{ h}$) compared to its usual timetable.

1. a) Clearly define what the unknown $v$ represents and set up a single equation relating the fixed distance, the two average speeds, and the time difference between both journeys.
2. b) Transform the previous relationship into a quadratic equation in standard form ($av^2+bv+c=0$), solve it, and determine the train's usual average speed and the normal journey duration.
3. c) The mathematical equation yields two real solutions ($v_1$ and $v_2$). Explain with reasoning why the negative root is discarded in this physical context.
4. d) Railway quality standards require that delays due to engineering works do not exceed $20\text{ minutes}$ ($\frac{1}{3}\text{ of an hour}$). Without needing to solve another complete quadratic equation, reason qualitatively whether the permissible speed reduction should be greater than or less than $50\text{ km/h}$ to comply with this standard.