Figure It Out: Identity 1C

1. Which is greater: $(a-b)^2$ or $(b-a)^2$?

Answer: They are equal. Proof: $(b-a) = -(a-b)$. Squaring a negative number makes it positive. $(b-a)^2 = [-(a-b)]^2 = (a-b)^2$

2. Express 100 as difference of two squares

We need $a^2 - b^2 = 100 \implies (a+b)(a-b) = 100$. Factors of 100 are: $(1,100), (2,50), (4,25), (5,20), (10,10)$. For $a$ and $b$ to be integers, factors must have the same parity (both even or both odd).

• Pair $(2, 50)$:
  • $a+b = 50, a-b = 2 \implies 2a = 52 \implies a=26, b=24$.
  • $26^2 - 24^2 = 676 - 576 = 100$.
• Pair $(10, 10)$:
  • $a=10, b=0$. $10^2 - 0^2 = 100$.

3. Calculating Squares

$406^2$ $(400+6)^2 = 160000 + 36 + 4800 = 164836$

$72^2$ $(70+2)^2 = 4900 + 4 + 280 = 5184$

$145^2$ $(150-5)^2 = 22500 + 25 - 1500 = 21025$

$1097^2$ $(1100-3)^2 = 1210000 + 9 - 6600 = 1203409$

$124^2$ $(125-1)^2 = 15625 + 1 - 250 = 15376$