5x^2+x+1=0

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Solution for 5x^2+x+1=0 equation:


Simplifying
5x2 + x + 1 = 0

Reorder the terms:
1 + x + 5x2 = 0

Solving
1 + x + 5x2 = 0

Solving for variable 'x'.

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

Divide each side by '5'.
0.2 + 0.2x + x2 = 0

Move the constant term to the right:

Add '-0.2' to each side of the equation.
0.2 + 0.2x + -0.2 + x2 = 0 + -0.2

Reorder the terms:
0.2 + -0.2 + 0.2x + x2 = 0 + -0.2

Combine like terms: 0.2 + -0.2 = 0.0
0.0 + 0.2x + x2 = 0 + -0.2
0.2x + x2 = 0 + -0.2

Combine like terms: 0 + -0.2 = -0.2
0.2x + x2 = -0.2

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

Add '0.25' to each side of the equation.
0.2x + 0.25 + x2 = -0.2 + 0.25

Reorder the terms:
0.25 + 0.2x + x2 = -0.2 + 0.25

Combine like terms: -0.2 + 0.25 = 0.05
0.25 + 0.2x + x2 = 0.05

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 0.05

Calculate the square root of the right side: 0.223606798

Break this problem into two subproblems by setting 
(x + 0.5) equal to 0.223606798 and -0.223606798.

Subproblem 1

x + 0.5 = 0.223606798 Simplifying x + 0.5 = 0.223606798 Reorder the terms: 0.5 + x = 0.223606798 Solving 0.5 + x = 0.223606798 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = 0.223606798 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 0.223606798 + -0.5 x = 0.223606798 + -0.5 Combine like terms: 0.223606798 + -0.5 = -0.276393202 x = -0.276393202 Simplifying x = -0.276393202

Subproblem 2

x + 0.5 = -0.223606798 Simplifying x + 0.5 = -0.223606798 Reorder the terms: 0.5 + x = -0.223606798 Solving 0.5 + x = -0.223606798 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = -0.223606798 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -0.223606798 + -0.5 x = -0.223606798 + -0.5 Combine like terms: -0.223606798 + -0.5 = -0.723606798 x = -0.723606798 Simplifying x = -0.723606798

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.276393202, -0.723606798}

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