A gardener wants a rectangular flower bed with an area of 96 m², with one side 4 m longer than the other. What are the dimensions of the bed? If you call the shorter side x, the longer side is x + 4 and the area is x · (x + 4) = 96. Here the unknown is multiplied by itself, so you get a quadratic equation.
A quadratic equation often has two roots, but the problem may have only one sensible answer. Let's see how to tell which.
Define the unknown. Write in words what x stands for (for example "x is the shorter side in metres").
Write the other quantities in terms of x.
Form the equation from the text and rearrange it into the form ax² + bx + c = 0.
Solve it, with the formula or by factorising.
Check the roots. A negative length, a negative number of people or a negative speed makes no sense, so throw that root away. Then answer in a full sentence.
Rectangles
Worked example 1
A flower bed has an area of 96 m² and one side is 4 m longer than the other.
x = shorter side (m), x + 4 = longer side.
x(x + 4) = 96, so x² + 4x − 96 = 0.
D = 16 + 384 = 400, x = (−4 ± 20) / 2, so x = 8 or x = −12.
−12 doesn't work, because a length can't be negative.
Answer: the bed measures 8 m by 12 m. Check: 8 · 12 = 96 ✓.
Consecutive numbers
Worked example 2
The product of two consecutive positive whole numbers is 156. What are the numbers?
n = the smaller number, n + 1 = the larger one.
n(n + 1) = 156, so n² + n − 156 = 0.
D = 1 + 624 = 625, n = (−1 ± 25) / 2, so n = 12 or n = −13.
−13 is not a positive whole number.
Answer: the numbers are 12 and 13.
A tournament where everyone plays everyone
If n teams each play every other team once, each team plays n − 1 matches. But the product n(n − 1) counts every match twice (once for each of the two teams), so the number of matches is n(n − 1) / 2.
Worked example 3
28 matches were played in a tournament. How many teams took part?
n(n − 1) / 2 = 28, so n² − n − 56 = 0.
D = 1 + 224 = 225, n = (1 ± 15) / 2, so n = 8 or n = −7.
Answer: 8 teams took part.
💡 In the same way, a polygon with n vertices has n(n − 3) / 2 diagonals: n − 3 diagonals start at each vertex, and each diagonal has been counted twice.
Right-angled triangles
Worked example 4
The two shorter sides of a right-angled triangle differ by 7 cm, and the hypotenuse is 13 cm. How long are the two shorter sides?
x = the shortest side, x + 7 = the other short side.
Pythagoras' theorem: x² + (x + 7)² = 13².
x² + x² + 14x + 49 = 169, so 2x² + 14x − 120 = 0; dividing by two gives x² + 7x − 60 = 0.
D = 49 + 240 = 289, x = (−7 ± 17) / 2, so x = 5 or x = −12.
Answer: the two shorter sides are 5 cm and 12 cm. Check: 25 + 144 = 169 ✓.
Speed and time
Journey time is distance divided by speed: t = s / v.
Worked example 5
A cyclist rode 60 km. If he had gone 5 km/h faster, he would have arrived an hour earlier. How fast did he ride?
v = his speed (km/h). His actual time is 60/v, the faster time 60/(v + 5).
The difference between the times is 1 hour: 60/v − 60/(v + 5) = 1.
Multiply by v(v + 5): 60(v + 5) − 60v = v(v + 5), so 300 = v² + 5v.
v² + 5v − 300 = 0, D = 25 + 1200 = 1225, v = (−5 ± 35) / 2, so v = 15 or v = −20.
Answer: the cyclist rode at 15 km/h. Check: 60/15 = 4 h, 60/20 = 3 h, a difference of 1 h ✓.
Common mistakes
Keeping a root that makes no sense. A negative length or a negative number of teams is not an answer.
An incomplete answer. If the question asks for the dimensions, give both sides, not just x.
An unclear unknown. If you don't write down what x is, it's easy to answer a different question.
Forgetting the units. The answer "8" isn't enough: it's 8 m.
In short
Define the unknown, write the other quantities in terms of it, form the equation, solve it.
A quadratic equation often has two roots, but the problem has only one sensible answer.
Throw away negative lengths, counts and speeds.
Check your result against the text of the problem and answer in a full sentence.
Try it yourself
The product of two consecutive even numbers is 168. What are the numbers?
A rectangle has a perimeter of 30 m and an area of 56 m². What are its dimensions?
A polygon has 20 diagonals. How many vertices does it have?
A path of the same width runs all the way round a 6 m × 10 m flower bed. The bed and the path together cover 140 m². How wide is the path?
Solutions:
n(n + 2) = 168, n² + 2n − 168 = 0, D = 676, n = 12 (−14 doesn't work). The numbers are 12 and 14.
a + b = 15, ab = 56: x² − 15x + 56 = 0, x = 7 or x = 8. The dimensions are 7 m and 8 m.
