4x(x)+1.41421356237x-1=0

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Solution for 4x(x)+1.41421356237x-1=0 equation:


Simplifying
4x(x) + 1.41421356237x + -1 = 0

Multiply x * x
4x2 + 1.41421356237x + -1 = 0

Reorder the terms:
-1 + 1.41421356237x + 4x2 = 0

Solving
-1 + 1.41421356237x + 4x2 = 0

Solving for variable 'x'.

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

Divide each side by '4'.
-0.25 + 0.3535533906x + x2 = 0

Move the constant term to the right:

Add '0.25' to each side of the equation.
-0.25 + 0.3535533906x + 0.25 + x2 = 0 + 0.25

Reorder the terms:
-0.25 + 0.25 + 0.3535533906x + x2 = 0 + 0.25

Combine like terms: -0.25 + 0.25 = 0.00
0.00 + 0.3535533906x + x2 = 0 + 0.25
0.3535533906x + x2 = 0 + 0.25

Combine like terms: 0 + 0.25 = 0.25
0.3535533906x + x2 = 0.25

The x term is 0.3535533906x.  Take half its coefficient (0.1767766953).
Square it (0.03125000000) and add it to both sides.

Add '0.03125000000' to each side of the equation.
0.3535533906x + 0.03125000000 + x2 = 0.25 + 0.03125000000

Reorder the terms:
0.03125000000 + 0.3535533906x + x2 = 0.25 + 0.03125000000

Combine like terms: 0.25 + 0.03125000000 = 0.28125
0.03125000000 + 0.3535533906x + x2 = 0.28125

Factor a perfect square on the left side:
(x + 0.1767766953)(x + 0.1767766953) = 0.28125

Calculate the square root of the right side: 0.530330086

Break this problem into two subproblems by setting 
(x + 0.1767766953) equal to 0.530330086 and -0.530330086.

Subproblem 1

x + 0.1767766953 = 0.530330086 Simplifying x + 0.1767766953 = 0.530330086 Reorder the terms: 0.1767766953 + x = 0.530330086 Solving 0.1767766953 + x = 0.530330086 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1767766953' to each side of the equation. 0.1767766953 + -0.1767766953 + x = 0.530330086 + -0.1767766953 Combine like terms: 0.1767766953 + -0.1767766953 = 0.0000000000 0.0000000000 + x = 0.530330086 + -0.1767766953 x = 0.530330086 + -0.1767766953 Combine like terms: 0.530330086 + -0.1767766953 = 0.3535533907 x = 0.3535533907 Simplifying x = 0.3535533907

Subproblem 2

x + 0.1767766953 = -0.530330086 Simplifying x + 0.1767766953 = -0.530330086 Reorder the terms: 0.1767766953 + x = -0.530330086 Solving 0.1767766953 + x = -0.530330086 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1767766953' to each side of the equation. 0.1767766953 + -0.1767766953 + x = -0.530330086 + -0.1767766953 Combine like terms: 0.1767766953 + -0.1767766953 = 0.0000000000 0.0000000000 + x = -0.530330086 + -0.1767766953 x = -0.530330086 + -0.1767766953 Combine like terms: -0.530330086 + -0.1767766953 = -0.7071067813 x = -0.7071067813 Simplifying x = -0.7071067813

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.3535533907, -0.7071067813}

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