Completing the square ... haven't done that in about nine or ten months so this will be rough xD
m^2 + 3m - 1 = 0
(I always put it in this form - something = 0 - because that is how it's the most familiar for me. It's not actually necessary :) Now, just to keep our algebraic muscles working, let's consider this as am^2 + bm + c, where a = 1, b = 3 and c = -1, so when I refer to the a term, b term and c term, that's what I mean. You should be used to considering quadratic equations in that general form.)
Divide the b-term by two and put it into the squared bit, and see what that comes out as:
(m + 3/2)^2 = m^2 + 3m + (3/2)^2
(3/2)^2 = (3^2)/(2^2) = 9/4
Which means that:
m^2 + 3m -1 = m^2 + 3m + 9/4 - 9/4 - 1
If you look at that: 9/4 - 9/4 = 0, yup? So both sides of that equation have the same value. But we now also have the (m + 3/2)^2 bit hanging around in there, in italics! Therefore:
m^2 + 3m - 1 = (m + 3/2)^2 - 9/4 - 1
M^2 + 3m - 1 = (m + 3/2)^2 - 13/4
(m + 3/2)^2 - 13/4 = 0
(m + 3/2)^2 = 13/4
M + 3/2 = +- SQRT (13/4)
Never forget that +/-, it's very important!
Once you understand the method, you quickly start to do something like this:
X^2 + bx + c = 0
Well, I know how that works, so I just automatically switch it to:
(x + b/2)^2 = -c - b^2/4
X + b/2 = +- SQRT (-c - b^2/4)
And solve from there.
Completing the square is how we get the quadratic formula, too :)
EDIT TO ADD:
Our final answer is that:
m + 3/2 = +- SQRT (13/4)
M1 = (SQRT13)/2 - 3/2
M2 = -(SQRT13)/2 - 3/2
Exact form would be as above: SQRT13 is a surd and hence irrational. Approximate form would be to stick that into your calculator and see what number it gives you.
If you believe everything you read, better not read.