Example 1 If a = 2021, b = 2022 and c = 2023, find the value of the expression a2 + b2 + c2 – ab – bc – ca.
Example 2 If the height of a triangle is decreased by 40% and its base is increased by 40%, what will be the percentage change in its area?
Of course, you should try/ solve these questions on your own and then check the solutions given.
Solution 1. If we put the data in a calculator, it will show as below:
Of course, the answer is 3. But how to do when a calculator is not available? To start with, look at the expression given in the question. Following probable questions may come to our mind:
(i) Is it possible to apply the formula of (a + b)2 or (a − b)2 anyway?
(ii) Is it possible to rearrange the terms?
(iii) Can we do something, as there is a symmetry in the expression?
Yes, rearrangement is possible. Recall that, first multiplying and then dividing by the same quantity will result in no change. Applying these concepts, we have
Solution 2. Let the height and base of the triangle ABC be h and b respectively (see the figure below).
Now, the height of the triangle is decreased by 40% but its base is increased by 40%.