The trick: someone says 'what's 7.5 squared?' and - mentally squaring in a flash - you say: 56.25.
Squaring halves is really easy if you know your times tables. Here's the method:
You can also use this to square any number that ends in 5. It's the same idea:
Why does this work? Well, it's the old 'difference of two squares' trick. Let me write it this way:
$$ (x + 0.5)(x - 0.5) = x^2 - 0.25$$
$$ (x + 0.5)(x - 0.5) + 0.25 = x^2$$
... and that's all there is to it!
You can use this trick backwards to get a better guess for square roots - for example, if you spot that 110 is $11 \times 10$, you can say that its square root must be a little less than 10.5, because $10.5^2 = 110.25$.