-4x²-3x+1 = 0

Make the leading term positive by multiplying by -1

4x²+3x-1 = 0

Get the -1 off the left side by adding 1 to both sides:

4x²+3x = 1

Divide every number through by 4

x²+x = 

x²+x = 

1. Multiply the coefficient  of x,  by , getting 
2. Square , getting  = 
3. Add  to both sides:

x²+x+ = +

The left side will now factor 
as a square: x²+x+ = (x+)(x+) = (x+)²
Combine the fractions on the right by getting a LCD of 64: + = + = 

(x+)² = 

take square roots of both sides:



(x+) = 

Add } to both sides:

x = 

Using the + :  x =  =  = 

Using the - :  x =  =  = -1

Solutions:  and -1

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It could have been done by factoring:

-4x²-3x+1 = 0

Make the leading term positive by multiplying by -1

4x²+3x-1 = 0

(4x-1)(x+1) = 0

4x-1 = 0;  x+1 = 0
  4x = 1     x = -1
   x = 

Edwin