109x^2+2x-1=0

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Solution for 109x^2+2x-1=0 equation:


Simplifying
109x2 + 2x + -1 = 0

Reorder the terms:
-1 + 2x + 109x2 = 0

Solving
-1 + 2x + 109x2 = 0

Solving for variable 'x'.

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

Divide each side by '109'.
-0.009174311927 + 0.01834862385x + x2 = 0

Move the constant term to the right:

Add '0.009174311927' to each side of the equation.
-0.009174311927 + 0.01834862385x + 0.009174311927 + x2 = 0 + 0.009174311927

Reorder the terms:
-0.009174311927 + 0.009174311927 + 0.01834862385x + x2 = 0 + 0.009174311927

Combine like terms: -0.009174311927 + 0.009174311927 = 0.000000000000
0.000000000000 + 0.01834862385x + x2 = 0 + 0.009174311927
0.01834862385x + x2 = 0 + 0.009174311927

Combine like terms: 0 + 0.009174311927 = 0.009174311927
0.01834862385x + x2 = 0.009174311927

The x term is 0.01834862385x.  Take half its coefficient (0.009174311925).
Square it (0.00008416799930) and add it to both sides.

Add '0.00008416799930' to each side of the equation.
0.01834862385x + 0.00008416799930 + x2 = 0.009174311927 + 0.00008416799930

Reorder the terms:
0.00008416799930 + 0.01834862385x + x2 = 0.009174311927 + 0.00008416799930

Combine like terms: 0.009174311927 + 0.00008416799930 = 0.0092584799263
0.00008416799930 + 0.01834862385x + x2 = 0.0092584799263

Factor a perfect square on the left side:
(x + 0.009174311925)(x + 0.009174311925) = 0.0092584799263

Calculate the square root of the right side: 0.096220995

Break this problem into two subproblems by setting 
(x + 0.009174311925) equal to 0.096220995 and -0.096220995.

Subproblem 1

x + 0.009174311925 = 0.096220995 Simplifying x + 0.009174311925 = 0.096220995 Reorder the terms: 0.009174311925 + x = 0.096220995 Solving 0.009174311925 + x = 0.096220995 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.009174311925' to each side of the equation. 0.009174311925 + -0.009174311925 + x = 0.096220995 + -0.009174311925 Combine like terms: 0.009174311925 + -0.009174311925 = 0.000000000000 0.000000000000 + x = 0.096220995 + -0.009174311925 x = 0.096220995 + -0.009174311925 Combine like terms: 0.096220995 + -0.009174311925 = 0.087046683075 x = 0.087046683075 Simplifying x = 0.087046683075

Subproblem 2

x + 0.009174311925 = -0.096220995 Simplifying x + 0.009174311925 = -0.096220995 Reorder the terms: 0.009174311925 + x = -0.096220995 Solving 0.009174311925 + x = -0.096220995 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.009174311925' to each side of the equation. 0.009174311925 + -0.009174311925 + x = -0.096220995 + -0.009174311925 Combine like terms: 0.009174311925 + -0.009174311925 = 0.000000000000 0.000000000000 + x = -0.096220995 + -0.009174311925 x = -0.096220995 + -0.009174311925 Combine like terms: -0.096220995 + -0.009174311925 = -0.105395306925 x = -0.105395306925 Simplifying x = -0.105395306925

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.087046683075, -0.105395306925}

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