Problems with quadratic equations
You already know how to solve $ax^2+bx+c=0$ using the formula or by factorising. The hard part of these problems is not the final calculation, but translating the wording into an equation, and then deciding what to do when there are two solutions.
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.
Numbers and sharing
When the problem talks about "consecutive numbers", it helps to name only one of them $x$ and write the rest in terms of it.
The product of two consecutive integers is $156$. What are these numbers?
The product of two consecutive integers is $156$. What are these numbers?
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…
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?
Geometry: areas and Pythagoras
In a right-angled triangle, the two shorter sides and the hypotenuse satisfy $a^2+b^2=c^2$. If one side is described as "x plus something", Pythagoras' equation is quadratic.
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.
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.
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.
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?
Ages and motion
Age problems combine "$n$ years ago" (subtraction) and "in $n$ years' time" (addition) inside a square or a product, which leads to a quadratic equation.
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 upwards. Its height $h$ (in metres) as a function of time $t$ (in seconds) is given by the mathematical expression $h = 40t - 5t^2$.…
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 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?
Be careful with the two solutions
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?
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,…
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?