4x^2=x+1

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


Simplifying
4x2 = x + 1

Reorder the terms:
4x2 = 1 + x

Solving
4x2 = 1 + x

Solving for variable 'x'.

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

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

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

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

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

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

Move the constant term to the right:

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

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

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

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

The x term is -0.25x.  Take half its coefficient (-0.125).
Square it (0.015625) and add it to both sides.

Add '0.015625' to each side of the equation.
-0.25x + 0.015625 + x2 = 0.25 + 0.015625

Reorder the terms:
0.015625 + -0.25x + x2 = 0.25 + 0.015625

Combine like terms: 0.25 + 0.015625 = 0.265625
0.015625 + -0.25x + x2 = 0.265625

Factor a perfect square on the left side:
(x + -0.125)(x + -0.125) = 0.265625

Calculate the square root of the right side: 0.515388203

Break this problem into two subproblems by setting 
(x + -0.125) equal to 0.515388203 and -0.515388203.

Subproblem 1

x + -0.125 = 0.515388203 Simplifying x + -0.125 = 0.515388203 Reorder the terms: -0.125 + x = 0.515388203 Solving -0.125 + x = 0.515388203 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.125' to each side of the equation. -0.125 + 0.125 + x = 0.515388203 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + x = 0.515388203 + 0.125 x = 0.515388203 + 0.125 Combine like terms: 0.515388203 + 0.125 = 0.640388203 x = 0.640388203 Simplifying x = 0.640388203

Subproblem 2

x + -0.125 = -0.515388203 Simplifying x + -0.125 = -0.515388203 Reorder the terms: -0.125 + x = -0.515388203 Solving -0.125 + x = -0.515388203 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.125' to each side of the equation. -0.125 + 0.125 + x = -0.515388203 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + x = -0.515388203 + 0.125 x = -0.515388203 + 0.125 Combine like terms: -0.515388203 + 0.125 = -0.390388203 x = -0.390388203 Simplifying x = -0.390388203

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.640388203, -0.390388203}

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