Here is another example. Compare the following identities from algebra: (a + b)2 = a2 + b2 + 2ab, (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca. Can you see that the first identity is a special case of the second one (put c = 0 in the second identity), and the second identity is a generalisation of the first one?
Answer: Put c = 0 in (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca: every term with c vanishes, leaving (a + b)2 = a2 + b2 + 2ab. So the first is a special case; the second, which works for any c, generalises it.
Step-by-step solution
Idea: A special case comes from a general result by adding an extra condition (here c = 0).
- Second identity: (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca. Put c = 0: LHS = (a + b)2; on the right c2, 2bc, 2ca all become 0.1 mark
- So (a + b)2 = a2 + b2 + 2ab, the first identity: a special case. The second holds for every value of c, not just 0, so it is a generalisation of the first.1 mark
Putting c = 0 in the three-term identity gives exactly (a + b)² = a² + b² + 2ab, so the first is a special case of the second, and the second generalises the first.
Answer to write in the exam
(a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
Put c = 0: (a + b)2 = a2 + b2 + 0 + 2ab + 0 + 0
∴ (a + b)2 = a2 + b2 + 2ab (special case)
Common mistakes that cost marks
- Putting c = 1 instead of 0; that gives a different identity.
- Mixing up the words: the special case is the smaller, more restricted result; the generalisation is the wider one.
How this can come in the exam
MCQ (1 mark)
Which value of b turns (a + b)2 = a2 + 2ab + b2 into (a − 1)2 = a2 − 2a + 1?
- 0
- 1
- −1
- 2
Show answer
(C) −1
Put b = −1: a2 + 2a(−1) + 1 = a2 − 2a + 1.
Try one yourself
Show that the formula for the area of a square, A = a2, is a special case of A = ab.
Show answer
A square is a rectangle with b = a; put b = a in A = ab to get A = a2.
More questions like this
- Try to work out why this method works. You will find that it is a geometrical translation of the formula (a + b2)2 − (a − b2)2 = ab.
- What procedure would you use to square a given triangle? Here, the task is to construct a square whose area is equal to the area of some given triangle. Think carefully. How would you proceed?
- Find the area of triangle ADE in the figure.
- The parallel sides of a trapezium are 40 cm and 20 cm. If its non-parallel sides are both equal, each being 26 cm, find the area of the trapezium.
- Find the area of a triangle, given that its sides are 8 cm and 11 cm long, and its perimeter is 32 cm.