Question

Why is -a² is always negative (a ≠ 0) and why is (-a)² is always positive?

Answer 1

`-a^2=(-1)\cdot a\cdot a`

Regardless of the sign of
`a`, we have
`a\cdot a=a^2\ge0` (never negative). But multiplying by -1 makes it negative.

On the other hand,


`(-a)^2=((-1)\cdot a)^2=(-1)^2\cdot a^2=1\cdot a^2=a^2`

which can never be negative for real
`a`.