9k^2-k-1=0

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Solution for 9k^2-k-1=0 equation:


Simplifying
9k2 + -1k + -1 = 0

Reorder the terms:
-1 + -1k + 9k2 = 0

Solving
-1 + -1k + 9k2 = 0

Solving for variable 'k'.

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

Divide each side by '9'.
-0.1111111111 + -0.1111111111k + k2 = 0

Move the constant term to the right:

Add '0.1111111111' to each side of the equation.
-0.1111111111 + -0.1111111111k + 0.1111111111 + k2 = 0 + 0.1111111111

Reorder the terms:
-0.1111111111 + 0.1111111111 + -0.1111111111k + k2 = 0 + 0.1111111111

Combine like terms: -0.1111111111 + 0.1111111111 = 0.0000000000
0.0000000000 + -0.1111111111k + k2 = 0 + 0.1111111111
-0.1111111111k + k2 = 0 + 0.1111111111

Combine like terms: 0 + 0.1111111111 = 0.1111111111
-0.1111111111k + k2 = 0.1111111111

The k term is -0.1111111111k.  Take half its coefficient (-0.05555555555).
Square it (0.003086419752) and add it to both sides.

Add '0.003086419752' to each side of the equation.
-0.1111111111k + 0.003086419752 + k2 = 0.1111111111 + 0.003086419752

Reorder the terms:
0.003086419752 + -0.1111111111k + k2 = 0.1111111111 + 0.003086419752

Combine like terms: 0.1111111111 + 0.003086419752 = 0.114197530852
0.003086419752 + -0.1111111111k + k2 = 0.114197530852

Factor a perfect square on the left side:
(k + -0.05555555555)(k + -0.05555555555) = 0.114197530852

Calculate the square root of the right side: 0.337931252

Break this problem into two subproblems by setting 
(k + -0.05555555555) equal to 0.337931252 and -0.337931252.

Subproblem 1

k + -0.05555555555 = 0.337931252 Simplifying k + -0.05555555555 = 0.337931252 Reorder the terms: -0.05555555555 + k = 0.337931252 Solving -0.05555555555 + k = 0.337931252 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.05555555555' to each side of the equation. -0.05555555555 + 0.05555555555 + k = 0.337931252 + 0.05555555555 Combine like terms: -0.05555555555 + 0.05555555555 = 0.00000000000 0.00000000000 + k = 0.337931252 + 0.05555555555 k = 0.337931252 + 0.05555555555 Combine like terms: 0.337931252 + 0.05555555555 = 0.39348680755 k = 0.39348680755 Simplifying k = 0.39348680755

Subproblem 2

k + -0.05555555555 = -0.337931252 Simplifying k + -0.05555555555 = -0.337931252 Reorder the terms: -0.05555555555 + k = -0.337931252 Solving -0.05555555555 + k = -0.337931252 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.05555555555' to each side of the equation. -0.05555555555 + 0.05555555555 + k = -0.337931252 + 0.05555555555 Combine like terms: -0.05555555555 + 0.05555555555 = 0.00000000000 0.00000000000 + k = -0.337931252 + 0.05555555555 k = -0.337931252 + 0.05555555555 Combine like terms: -0.337931252 + 0.05555555555 = -0.28237569645 k = -0.28237569645 Simplifying k = -0.28237569645

Solution

The solution to the problem is based on the solutions from the subproblems. k = {0.39348680755, -0.28237569645}

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