4x-2x^2=1

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


Simplifying
4x + -2x2 = 1

Solving
4x + -2x2 = 1

Solving for variable 'x'.

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

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

Begin completing the square.  Divide all terms by
-2 the coefficient of the squared term: 

Divide each side by '-2'.
0.5 + -2x + x2 = 0

Move the constant term to the right:

Add '-0.5' to each side of the equation.
0.5 + -2x + -0.5 + x2 = 0 + -0.5

Reorder the terms:
0.5 + -0.5 + -2x + x2 = 0 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + -2x + x2 = 0 + -0.5
-2x + x2 = 0 + -0.5

Combine like terms: 0 + -0.5 = -0.5
-2x + x2 = -0.5

The x term is -2x.  Take half its coefficient (-1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
-2x + 1 + x2 = -0.5 + 1

Reorder the terms:
1 + -2x + x2 = -0.5 + 1

Combine like terms: -0.5 + 1 = 0.5
1 + -2x + x2 = 0.5

Factor a perfect square on the left side:
(x + -1)(x + -1) = 0.5

Calculate the square root of the right side: 0.707106781

Break this problem into two subproblems by setting 
(x + -1) equal to 0.707106781 and -0.707106781.

Subproblem 1

x + -1 = 0.707106781 Simplifying x + -1 = 0.707106781 Reorder the terms: -1 + x = 0.707106781 Solving -1 + x = 0.707106781 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + x = 0.707106781 + 1 Combine like terms: -1 + 1 = 0 0 + x = 0.707106781 + 1 x = 0.707106781 + 1 Combine like terms: 0.707106781 + 1 = 1.707106781 x = 1.707106781 Simplifying x = 1.707106781

Subproblem 2

x + -1 = -0.707106781 Simplifying x + -1 = -0.707106781 Reorder the terms: -1 + x = -0.707106781 Solving -1 + x = -0.707106781 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + x = -0.707106781 + 1 Combine like terms: -1 + 1 = 0 0 + x = -0.707106781 + 1 x = -0.707106781 + 1 Combine like terms: -0.707106781 + 1 = 0.292893219 x = 0.292893219 Simplifying x = 0.292893219

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.707106781, 0.292893219}

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