Hey there!
What your teacher is essentially trying to tell you is to complete the square such that the equation of the circle is in the form \((x-h)^2 + (y-k)^2 = r^2\), where the centre of the circle is (h, k) and the radius is r. When we expand this, we have that \(x^2-2hx+h^2+y^2-2yk+k^2 = r^2\), and from this, we can surmise what the values of h and k are from the 'linear equation' (even though it's not technically linear, it's more unfactorised, and not even that
). When we complete the square, we ideally add and subtract constants to keep the equation the same, while also allowing us to complete the square. For example, in 3a), we have that:
Notice how we added and subtracted four to complete the square, then moved the four we subtracted to the other side so we were able to manipulate the equation of the circle into the form \((x-h)^2 + (y-k)^2 = r^2\).
For 3b) a similar thing happens:
Basically, it's all about manipulating the equation into a form you can recognise, using methods that keep the equation the same, in this case adding and subtracting the same number. In other cases, you'll need to multiply by a number and its reciprocal or use some identity.
Have a go at the rest yourself! Hope this helps