# Where's the problem in this equation? Resulting in $4 = 5$

I just saw this equation and I can't find out where's the problem:

$$25-45 = 16-36$$

$$25- 2 \cdot 5 \cdot \frac{9}{2} = 16- 2\cdot4\cdot\frac{9}{2}$$

$$25 - 2\cdot 5\cdot \frac{9}{2} + \frac{81}{4} = 16 - 2\cdot 4 \cdot \frac{9}{2} + \frac{81}{4}$$

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

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

• you could try to solve the left and right side on each line and see where it starts to differ – ratchet freak Jul 14 '12 at 15:00
• Besides what has been said in the answers, there's actually another mistake: your brackets in the third line are not right. – Raskolnikov Jul 14 '12 at 15:17
• The brackets were made wrong by an edit from Joe L. I have submitted a correction. – user12861 Jul 14 '12 at 15:46
• Related, same fallacy. – Daniel Fischer Nov 19 '16 at 13:03
• Not the same, but rather similar approach: $2+2 = 5$? error in proof – Martin Sleziak Nov 19 '16 at 13:20

$a^2 = b^2$ does not imply that $a = b$.
• It only implies $|a|=|b|$, i.e. $a=\pm b$. – Martin Sleziak Jul 14 '12 at 10:48
$5-9/2$ is positive, $4-9/2$ is negative. They are not equal, although their squares are equal.