Problems with quadratic equations
Ximena and her friends are pooling money for a gift. If two more friends join, each person pays less. How many were there at the start? You already know how to solve $ax^2+bx+c=0$ with the quadratic formula. The hard part is translating the problem statement into an equation and deciding what to do when you get two solutions. At the end of the page we will solve the gift problem.
Setting up the problem
Find a positive number such that three times its square is exactly equal to twelve times the number itself.
Find a positive number such that three times its square is exactly equal to twelve times the number itself.
Consecutive numbers and sharing costs
When the problem talks about consecutive numbers, name only one of them $x$ and write the others in terms of it: $x$, $x+1$ (consecutive integers); $x$, $x+2$ (consecutive even or odd numbers).
The product of two consecutive integers is $156$. What are these numbers?
The product of two consecutive integers is $156$. What are these numbers?
The product of two consecutive positive even numbers is $288$. Find these numbers.
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.
The sum of the squares of two consecutive odd natural numbers is $130$. Determine which numbers these are.
A group of friends decides to buy a wedding gift costing $2\,400$ pesos, sharing the cost equally. At the last minute, two more friends join the group, which means each…
A group of friends decides to buy a wedding gift costing $2\,400$ pesos, sharing the cost equally. At the last minute, two more friends join the group, which means each of the previous members pays $60$ pesos less than initially planned. How many friends were in the initial group?
Geometry: areas, volumes, and the Pythagorean theorem
In a right triangle, $a^2+b^2=c^2$. If one side is "$x$ plus something", the Pythagorean theorem gives a quadratic equation. The same thing happens with areas.
The base of a rectangular plot measures $5$ meters more than its height. If its total area is $150 \text{ m}^2$, calculate its dimensions.
The base of a rectangular plot measures $5$ meters more than its height. If its total area is $150 \text{ m}^2$, calculate its dimensions.
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.
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.
If the side of a ceramic square tile is increased by $4 \text{ cm}$, its area quadruples. How long was the side initially?
If the side of a ceramic square tile is increased by $4 \text{ cm}$, its area quadruples. How long was the side initially?
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…
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…
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 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…
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.
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 $x$…
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.
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…
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 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…
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.
Ages and motion
Age problems combine "$n$ years ago" (subtraction) and "in $n$ years" (addition) inside a square or a product. If someone is $x$ years old today, 2 years ago they were $x-2$ and in 3 years they will be $x+3$.
A projectile that goes up and comes down also gives a quadratic. Look at the graph: at which heights does it pass twice?
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?
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?
A projectile is launched vertically upward. Its height $h$ (in meters) as a function of time $t$ (in seconds) is given by the mathematical expression $h = 40t - 5t^2$.…
A projectile is launched vertically upward. Its height $h$ (in meters) 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 $75 \text{ meters}$?
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…
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 a track and field 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…
In a track and field 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 meters and $y$ is the height in meters. At what horizontal distance will it hit the ground?
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…
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?
Money, times, and speeds
Many problems about prices, work, and trips have the same structure as the gift: a total quantity divided by an unknown. The useful relationships:
-
Total revenue = price per ticket × number of tickets.
-
Distance = speed × time, so time = distance ÷ speed.
-
Amount with compound interest over 2 years: $C(1+i)^2$.
-
Two faucets (or pipes) that fill together in $T$ hours, and alone in $a$ and $b$ hours: $\dfrac1a+\dfrac1b=\dfrac1T$.
The cost price of a batch of T-shirts is $6\,000$ pesos. The seller decides to keep $5$ T-shirts for their family and sell the rest for $40$ pesos more than each cost.…
The cost price of a batch of T-shirts is $6\,000$ pesos. The seller decides to keep $5$ T-shirts for their family and sell the rest for $40$ pesos more than each cost. If, after selling them all, they make a net profit of $1\,200$ pesos (that is, receiving a total of $7\,200$ pesos in revenue), how many T-shirts were bought initially?
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.…
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?
An investor invests a principal of $100\,000$ pesos at a certain annual compound interest rate. After two years, the accumulated amount is $108\,160$ pesos. What was the…
An investor invests a principal of $100\,000$ pesos at a certain annual compound interest rate. After two years, the accumulated amount is $108\,160$ pesos. 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…
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?
A theater company rents a hall for $16\,000$ pesos for a performance. To cover expenses, they decide to set a ticket price. On the day of the performance, $40$ more…
A theater company rents a hall for $16\,000$ pesos 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 organizer decides to lower the ticket price by $20$ pesos, raising exactly the necessary $16\,000$ pesos. How many audience members actually attended the performance?