Challenge Problem: Spot the Error in the Proof

Question

Vocaloid · Answer

-20 = -20

re-write this as

16 - 36 = 25 - 45

re-write 16 and 25 as perfect squares, and 36 and 45 as products of two factors

$4^{2} - 4*9 = 5^{2} - 5*9 $

add 81/4 to both sides

$4^{2} - 4*9 + \frac{ 81 }{ 4 } = 5^{2} - 5*9 +  \frac{ 81 }{ 4 } $

re-write 81/4 as (9/2)^2 and re-write 4*9 as 2*(9/2)*4 and 5*9 as 2(9/2)*5

$4^{2} - 2*(\frac{ 9 }{ 2 })*4 + (\frac{ 9 }{ 2 })^{2} = 5^{2} - 2*(\frac{ 9 }{ 2 })*5 +  (\frac{ 9 }{ 2 })^{2} $

you can factor this, using the rule that a^2 -2ab + b^2 = (a-b)^2

(4 - $\frac{ 9 }{ 2 }$)^{2} = (5 - $\frac{ 9 }{ 2 }$)^{2}

take the square root of both sides

(4 - $\frac{ 9 }{ 2 }$) = (5 - $\frac{ 9 }{ 2 }$)

add 9/2 to both sides

4 = 5

subtract 4 from both sides

0 = 1

sillybilly123 · Answer

$a^2 = b^2 \implies  a = b $  ?!

Vocaloid · Answer

well done, that's the error ^^

sillybilly123 · Answer

happy Xmas etc :)