Answer:
The original 2x^2 - 4x = -3 becomes 2[ (x - 1)^2 - 1 = -3.
Explanation:
Start with 2x^2-4x=-3. Rewrite this as 2x^2-4x = -3, and then as
2(x^2 - 2x ) = -3.
Next, "complete the square" of x^2 - 2x: Take half of the coefficient (-2) of x (which comes out to -1). Square this result (obtaining (-1)^2, or just 1. Add this 1 to x^2 - 2x, and then subtract the same quantity (1): x^2 - 2x + 1 - 1.
Now rewrite the perfect square x^2 - 2x + 1 as (x - 1)^2, and then subtract that -1 (above):
(x - 1)^2 - 1.
Substituting this for x^2 - 2x in 2(x^2 - 2x ) = -3, we get:
2 [ (x - 1)^2 - 1 ] = -3, or
2(x - 1)^2 - 2 = - 3, or
2(x -1)^2 = -1
We weren't asked to solve this equation, but may as well do so:
Divide both sides by 2, obtaining (x - 1)^2 = -1/2
Normally, we'd find the square root of both sides, but here we cannot find the square root of the negative number -1/2, unless complex roots are acceptable.