First, get the value in front of x² by itself by dividing everything by the coefficient before it, yielding:
x²  2x = 3/2
Now you have to add the value of (b/2)² to both sides, where b is the values in front of the x term, so:
x²  2x + 1 = 3/2 + 1
Now you can factor the left to (x + b/2)²:
(x  1)² = 5/2
Take the square root of both sides, keeping in mind that a square root produces a positive and negative result:
x  1 = ±√(5/2)
Solve for x:
x = 1 ± √(5/2)
Hope that helps.

EDIT: Your book probably felt like being lazy and using the quadratic formula to get a quick solution. If you let:
x^2  2x  (3/2) = 0
Then...
x = [2 ± √(4  4 *1 * 3/2)]/2
x = [2 ± √(4 + 6)]/2
x = [2 ± √(10)]/2
Which can be simplified to...
x = 1 ± √(10)/2
x = 1 ± √(10/4)
x = 1 ± √(5/2)
That's the same thing.
Also, neat trick Robert, shame it doesn't work...
x²  2x = 3/2
Now you have to add the value of (b/2)² to both sides, where b is the values in front of the x term, so:
x²  2x + 1 = 3/2 + 1
Now you can factor the left to (x + b/2)²:
(x  1)² = 5/2
Take the square root of both sides, keeping in mind that a square root produces a positive and negative result:
x  1 = ±√(5/2)
Solve for x:
x = 1 ± √(5/2)
Hope that helps.

EDIT: Your book probably felt like being lazy and using the quadratic formula to get a quick solution. If you let:
x^2  2x  (3/2) = 0
Then...
x = [2 ± √(4  4 *1 * 3/2)]/2
x = [2 ± √(4 + 6)]/2
x = [2 ± √(10)]/2
Which can be simplified to...
x = 1 ± √(10)/2
x = 1 ± √(10/4)
x = 1 ± √(5/2)
That's the same thing.
Also, neat trick Robert, shame it doesn't work...

Glad someone provided a clear answer. Sorry about that, I should have explained it better :)
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You were taught to take 1/2 of "b", then square that result and add it to both sides.
In this problwm you have to divide by "a" first. Notice this creates a fraction with the 3.
Here is a method not taught anymore. NO FRACTIONS till the last step.
2x²  4x  3 = 0
ax^2+bx+c=0
In this problem a=2 and b=3
Here is what you do:
Multiply every term by 4a..........4a=....4(2)=..............8
Add b^2....................................b…
8(2x²  4x  3 = 0)
16x^232x24=0
16x^232x=+24
16x^232x+9=+24+9.....You just created a Perfect Square Trinomial which is the pupose
......................................… Completing The Square................Now factor the left side
In this problwm you have to divide by "a" first. Notice this creates a fraction with the 3.
Here is a method not taught anymore. NO FRACTIONS till the last step.
2x²  4x  3 = 0
ax^2+bx+c=0
In this problem a=2 and b=3
Here is what you do:
Multiply every term by 4a..........4a=....4(2)=..............8
Add b^2....................................b…
8(2x²  4x  3 = 0)
16x^232x24=0
16x^232x=+24
16x^232x+9=+24+9.....You just created a Perfect Square Trinomial which is the pupose
......................................… Completing The Square................Now factor the left side
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