58 Azim Premji University At Right Angles, July 2022
Student Corner
Day 1
Our maths teacher, Krittika Ma’am, gave us a very interesting topic to work on. She gave us a few patterns like this. . ..
1. 82−72=15 2. 42−32=7 3. 132−122 =25 4. 102−92=19. . .
and told us to find a pattern between the LHS and RHS. We quickly figured out that we just need to add up the numbers to get to RHS. She pointed out that what we had on the LHS was a difference of squares of consecutive numbers (where the gap between the numbers is 1).
Day 2
Ma’am told us to work with a difference of squares where the gap is 2 instead of 1 and then with 3, and so on. We had to find a rule which holds true for all the cases. So, we all put on our thinking caps and came up with two different ideas.
Observations by the class.The difference of two squares is always the product of their sum and their difference.
∴a2−b2= (a+b)(a−b)
1
A New Way of Looking at the Difference-of-
Two-Squares Identity
BEDANTO
BHATTACHARJEE &
RIDDHI SARKAR
Azim Premji University At Right Angles, July 2022 59
Example:
(i) 82−62=(8−6)(8+6)=2×14=28
(ii) 132−122=(13−12)(13+12)=1×25=25 (iii) 42−22=(4−2)(4+2)=2×6=12
(iv) 102−52=(10−5)(10+5)=5×15=75 Observations by Bedanto and Riddhi (for examples 1 & 3)
Bedanto Riddhi
(i) 82−62=2×(8−6)×(8−1)= 2×2×7=28
(iii) 42−22=2×(4−2)×(4−1)= 2×2×3=12
(i) 82−62=2×(8−6)×(6+1)= 2×2×7=28
(iii) 42−22=2×(4−2)×(2+1)= 2×2×3=12
We then understood that we were arriving at the same product.
Now we tried our theory with the other examples, but we saw that they were not working.
For eg- 132−122̸=2×(13−12)×(13−1); Also 132−122̸=2×(13−12)×(12+1);
∴We had to generalize it. We observed that in example (i) 8−1=6+1 and in (iii) 4−1=2+1 We realised, 82−62=2×(8−6)×(6+1)=2×difference×mean
And 42−22=2×(4−2)×(4−1)=2×difference×mean
After we had a generalized form, we tested our theory with other examples-
∴ (ii) 132−122 =2×difference×mean=2×(13−12)×(12.5) =25 (iv) 102−52 =2×difference×mean=2×(10−5)×(7.5) =75
So, our conclusion was that- For, the difference of two squares, the difference is always twice the product of their difference and their mean.
∴ a2−b2 =2×(a−b)×(mean ofaandb) We had a question to answer! How are the two methods related?
One of our classmates, Anurag, figured it out. . .Identity the class arrived at:
a2−b2 = (a+b)(a−b) Our method-
a2−b2= 2×(a−b)×(mean ofaandb) =2×(a−b)
(a+b 2
)
= (a+b)(a−b)
BEDANTO BHATTACHARJEE (8B) & RIDDHI SARKAR (8B) The Future Foundation School, Kolkata
Email: [email protected]
