The student council wants to turn a corner of the school lot into a rectangular vegetable garden with an area of $84\text{ ft}^2$. The long side must be $5$ ft longer than the short side. How long is each side? Guessing could take a while. A quadratic equation answers it in a minute.
You already know how to solve $ax^2+bx+c=0$ by factoring, completing the square, or the quadratic formula. The hard part of word problems is not the final calculation. It is translating the words into an equation, then deciding what to do when you get two answers.
The three-step plan
Once you have your solutions, ask what the variable represents. Here is a quick guide.
What $x$ represents
Reject a solution when it is...
Length, width, radius, speed, time
negative or zero
Age
negative, or makes someone's age impossible
People, shirts, tickets
negative or not a whole number
Integers, temperatures, coordinates
usually fine as negative, so read the problem
Anything larger than the total available
too big (for example, cutting more than the whole side)
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Find a positive number such that three times its square is exactly equal to twelve times that same number.
Find a positive number such that three times its square is exactly equal to twelve times that same number.
For a rectangle, $A=\ell w$. For a square, $A=s^2$. When the problem relates one side to the other ("5 ft longer than...", "twice the width minus 1"), pick one side as $x$ and write the other side using $x$.
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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.
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.
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…
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.
A border of uniform width $x$ around a rectangle adds $x$ on each side. So each dimension grows by $2x$, not $x$. The area of the border is
$$\text{outer area}-\text{inner area}.$$
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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…
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?
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…
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 $x$.
In a right triangle with legs $a$ and $b$ and hypotenuse $c$ (the longest side, opposite the right angle):
$$a^2+b^2=c^2.$$
If a leg is described as "$x$ plus something," the equation becomes quadratic.
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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.
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.
Some problems need a formula you may not have used in a while.
When you cut equal squares from the corners of a sheet and fold up the sides, each dimension of the base loses $2x$ (one cut at each end), and the height of the box is $x$.
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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…
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 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…
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.
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…
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.
Age problems mix "years ago" (subtract) with "years from now" (add), often inside a square or a product. Make a small table first.
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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?
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?
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…
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?
These problems have the unknown in a denominator. Multiply every term by the least common denominator to clear the fractions. A quadratic equation appears.
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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…
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?
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…
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 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…
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?
For motion: $d = r\cdot t$, so $t=\dfrac{d}{r}$. When two trips cover the same distance at different speeds, write each time as $\dfrac{d}{r}$ and relate the times.
When two objects move along perpendicular paths, the distance between them is the hypotenuse of a right triangle, and each leg is $\text{rate}\times\text{time}$.
Work problems. If a job takes $h$ hours alone, you finish $\dfrac1h$ of the job each hour. Working together, the fractions add: $\dfrac1{h_1}+\dfrac1{h_2}=\dfrac1{h_{\text{together}}}$.
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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…
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?
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…
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?
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?
If you put a principal $P$ in an account that pays an annual rate $r$ (as a decimal), compounded once a year, the balance after $t$ years is
$$A=P(1+r)^t.$$
After $2$ years, $A=P(1+r)^2$. That is a quadratic in $r$, so finding an unknown rate leads to a quadratic equation.
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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…
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?
The path of a thrown or launched object is a parabola. Two questions come up all the time:
"When is it at height $k$?" Set the height formula equal to $k$.
"When does it hit the ground?" Set the height equal to $0$.
The equation $-16t^2+48t=32$ has two solutions because a horizontal line at $32$ ft cuts the parabola twice: once on the way up and once on the way down. (Here the graph uses $x$ for time.)
Sometimes the coefficients are decimals and the leading coefficient is negative. Multiply by a power of $10$ to clear decimals, then by $-1$ to make $a$ positive.
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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…
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 $75 \text{ meters}$?
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 feet and $y$ is the height in feet. At what horizontal distance will it touch the ground?
Back to the start: the garden is $7$ ft by $12$ ft. You wrote one quadratic equation, solved it, and checked both answers against the words of the problem.
Summary
Let's practice! 8 medium exercises in this section