Subtracting Quarters of Squares Equals Multiply?!

Solution 1:

Solution 2:

Expanding the squared terms gives \begin{equation} \frac{(a + b)^2}{4} - \frac{(a - b)^2}{4} = \frac{a^2 + 2ab + b^2}{4} - \frac{a^2 - 2ab + b^2}{4} = \frac{4ab}{4} = ab. \end{equation}

Solution 3:

In addition to the direct derivations already shown, your magical equation is closely related to the formula $$x^2 - y^2 = (x + y)(x - y).$$ Just set $a = x + y$ and $b = x - y.$ Then $\frac{a+b}2 = x$ and $\frac{a-b}2 = y,$ so $x^2 = \frac{(a+b)^2}4$ and $y^2 = \frac{(a-b)^2}4.$ Use these facts to replace $x^2,\ y^2,\ x + y,$ and $x - y$ in the equation above and you will have derived your magical equation in $a$ and $b.$