Exercises: Problems with quadratic equations

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

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

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

If we subtract three times the age Lucy was $2$ years ago from the square of the age she will be in $3$ years, we obtain $55$. How old is Lucy currently?

In a right 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 positive integers is $130$. Determine what these numbers are.

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

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

To make a kite, a rhombus-shaped fabric is needed. It is known that the longer diagonal measures $4 \text{ cm}$ more than the shorter 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 gift that costs $120$ dollars, to be paid in equal parts. At the last minute, two more friends join the group, which means that each of the previous ones has to pay $3$ dollars less than initially planned. How many friends made up the initial group?

A projectile is launched vertically upward. Its height $h$ (in feet) 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 cost price of a batch of T-shirts is $300$ dollars. The seller decides to keep $5$ T-shirts for their family and sell the rest for $2$ dollars more than each one cost them. If after selling them all they make a net profit of $60$ dollars (that is, they take in $360$ dollars in total), how many T-shirts did they buy initially?

A technology 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. What are their current ages?

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 feet and $y$ is the height in feet. At what horizontal distance will it touch the ground?

A rectangular cardboard sheet $30 \text{ cm}$ long by $20 \text{ cm}$ wide is available. You want to build an open-top box by cutting a square of side $x$ from each corner and folding the flaps upward. If the area of the base 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 at the same time from the same intersection along two perpendicular roads. One travels $10 \text{ km/h}$ faster than the other. After $2$ hours, the straight-line distance separating them is $100 \text{ km}$. At what speed is each one traveling?

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

A cyclist travels from Town A to Town B, which are $60 \text{ km}$ apart. On the way back, due to fatigue, their average speed is $5 \text{ km/h}$ less than on the outward journey, so they take $1$ hour more to complete the same route. At what speed did they travel on the outward journey?

A craftsman 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 thickness of $x \text{ cm}$ at the top and bottom, but on the sides the thickness 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 theater company rents a hall for $\$800$ for a performance. To cover the costs, they decide to set a price for the ticket. On the day of the performance, $40$ more spectators attend than expected, so the organizer decides to lower the ticket price by $\$1$, collecting exactly the $\$800$ needed. How many spectators finally 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 has been used to manufacture the can (including the two circular lids and the lateral face), calculate the radius of the base.