16x^2+x+1=0

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Solution for 16x^2+x+1=0 equation:


Simplifying
16x2 + x + 1 = 0

Reorder the terms:
1 + x + 16x2 = 0

Solving
1 + x + 16x2 = 0

Solving for variable 'x'.

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

Divide each side by '16'.
0.0625 + 0.0625x + x2 = 0

Move the constant term to the right:

Add '-0.0625' to each side of the equation.
0.0625 + 0.0625x + -0.0625 + x2 = 0 + -0.0625

Reorder the terms:
0.0625 + -0.0625 + 0.0625x + x2 = 0 + -0.0625

Combine like terms: 0.0625 + -0.0625 = 0.0000
0.0000 + 0.0625x + x2 = 0 + -0.0625
0.0625x + x2 = 0 + -0.0625

Combine like terms: 0 + -0.0625 = -0.0625
0.0625x + x2 = -0.0625

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.
0.0625x + 0.25 + x2 = -0.0625 + 0.25

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

Combine like terms: -0.0625 + 0.25 = 0.1875
0.25 + 0.0625x + x2 = 0.1875

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 0.1875

Calculate the square root of the right side: 0.433012702

Break this problem into two subproblems by setting 
(x + 0.5) equal to 0.433012702 and -0.433012702.

Subproblem 1

x + 0.5 = 0.433012702 Simplifying x + 0.5 = 0.433012702 Reorder the terms: 0.5 + x = 0.433012702 Solving 0.5 + x = 0.433012702 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 = 0.433012702 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 0.433012702 + -0.5 x = 0.433012702 + -0.5 Combine like terms: 0.433012702 + -0.5 = -0.066987298 x = -0.066987298 Simplifying x = -0.066987298

Subproblem 2

x + 0.5 = -0.433012702 Simplifying x + 0.5 = -0.433012702 Reorder the terms: 0.5 + x = -0.433012702 Solving 0.5 + x = -0.433012702 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 = -0.433012702 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -0.433012702 + -0.5 x = -0.433012702 + -0.5 Combine like terms: -0.433012702 + -0.5 = -0.933012702 x = -0.933012702 Simplifying x = -0.933012702

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.066987298, -0.933012702}

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