0=4x^2-2x-1

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Solution for 0=4x^2-2x-1 equation:


Simplifying
0 = 4x2 + -2x + -1

Reorder the terms:
0 = -1 + -2x + 4x2

Solving
0 = -1 + -2x + 4x2

Solving for variable 'x'.

Combine like terms: 0 + 1 = 1
1 + 2x + -4x2 = -1 + -2x + 4x2 + 1 + 2x + -4x2

Reorder the terms:
1 + 2x + -4x2 = -1 + 1 + -2x + 2x + 4x2 + -4x2

Combine like terms: -1 + 1 = 0
1 + 2x + -4x2 = 0 + -2x + 2x + 4x2 + -4x2
1 + 2x + -4x2 = -2x + 2x + 4x2 + -4x2

Combine like terms: -2x + 2x = 0
1 + 2x + -4x2 = 0 + 4x2 + -4x2
1 + 2x + -4x2 = 4x2 + -4x2

Combine like terms: 4x2 + -4x2 = 0
1 + 2x + -4x2 = 0

Begin completing the square.  Divide all terms by
-4 the coefficient of the squared term: 

Divide each side by '-4'.
-0.25 + -0.5x + x2 = 0

Move the constant term to the right:

Add '0.25' to each side of the equation.
-0.25 + -0.5x + 0.25 + x2 = 0 + 0.25

Reorder the terms:
-0.25 + 0.25 + -0.5x + x2 = 0 + 0.25

Combine like terms: -0.25 + 0.25 = 0.00
0.00 + -0.5x + x2 = 0 + 0.25
-0.5x + x2 = 0 + 0.25

Combine like terms: 0 + 0.25 = 0.25
-0.5x + x2 = 0.25

The x term is -0.5x.  Take half its coefficient (-0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
-0.5x + 0.0625 + x2 = 0.25 + 0.0625

Reorder the terms:
0.0625 + -0.5x + x2 = 0.25 + 0.0625

Combine like terms: 0.25 + 0.0625 = 0.3125
0.0625 + -0.5x + x2 = 0.3125

Factor a perfect square on the left side:
(x + -0.25)(x + -0.25) = 0.3125

Calculate the square root of the right side: 0.559016994

Break this problem into two subproblems by setting 
(x + -0.25) equal to 0.559016994 and -0.559016994.

Subproblem 1

x + -0.25 = 0.559016994 Simplifying x + -0.25 = 0.559016994 Reorder the terms: -0.25 + x = 0.559016994 Solving -0.25 + x = 0.559016994 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + x = 0.559016994 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + x = 0.559016994 + 0.25 x = 0.559016994 + 0.25 Combine like terms: 0.559016994 + 0.25 = 0.809016994 x = 0.809016994 Simplifying x = 0.809016994

Subproblem 2

x + -0.25 = -0.559016994 Simplifying x + -0.25 = -0.559016994 Reorder the terms: -0.25 + x = -0.559016994 Solving -0.25 + x = -0.559016994 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + x = -0.559016994 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + x = -0.559016994 + 0.25 x = -0.559016994 + 0.25 Combine like terms: -0.559016994 + 0.25 = -0.309016994 x = -0.309016994 Simplifying x = -0.309016994

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.809016994, -0.309016994}

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