10x^2-x-1=0

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Solution for 10x^2-x-1=0 equation:


Simplifying
10x2 + -1x + -1 = 0

Reorder the terms:
-1 + -1x + 10x2 = 0

Solving
-1 + -1x + 10x2 = 0

Solving for variable 'x'.

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

Divide each side by '10'.
-0.1 + -0.1x + x2 = 0

Move the constant term to the right:

Add '0.1' to each side of the equation.
-0.1 + -0.1x + 0.1 + x2 = 0 + 0.1

Reorder the terms:
-0.1 + 0.1 + -0.1x + x2 = 0 + 0.1

Combine like terms: -0.1 + 0.1 = 0.0
0.0 + -0.1x + x2 = 0 + 0.1
-0.1x + x2 = 0 + 0.1

Combine like terms: 0 + 0.1 = 0.1
-0.1x + x2 = 0.1

The x term is -0.1x.  Take half its coefficient (-0.05).
Square it (0.0025) and add it to both sides.

Add '0.0025' to each side of the equation.
-0.1x + 0.0025 + x2 = 0.1 + 0.0025

Reorder the terms:
0.0025 + -0.1x + x2 = 0.1 + 0.0025

Combine like terms: 0.1 + 0.0025 = 0.1025
0.0025 + -0.1x + x2 = 0.1025

Factor a perfect square on the left side:
(x + -0.05)(x + -0.05) = 0.1025

Calculate the square root of the right side: 0.320156212

Break this problem into two subproblems by setting 
(x + -0.05) equal to 0.320156212 and -0.320156212.

Subproblem 1

x + -0.05 = 0.320156212 Simplifying x + -0.05 = 0.320156212 Reorder the terms: -0.05 + x = 0.320156212 Solving -0.05 + x = 0.320156212 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.05' to each side of the equation. -0.05 + 0.05 + x = 0.320156212 + 0.05 Combine like terms: -0.05 + 0.05 = 0.00 0.00 + x = 0.320156212 + 0.05 x = 0.320156212 + 0.05 Combine like terms: 0.320156212 + 0.05 = 0.370156212 x = 0.370156212 Simplifying x = 0.370156212

Subproblem 2

x + -0.05 = -0.320156212 Simplifying x + -0.05 = -0.320156212 Reorder the terms: -0.05 + x = -0.320156212 Solving -0.05 + x = -0.320156212 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.05' to each side of the equation. -0.05 + 0.05 + x = -0.320156212 + 0.05 Combine like terms: -0.05 + 0.05 = 0.00 0.00 + x = -0.320156212 + 0.05 x = -0.320156212 + 0.05 Combine like terms: -0.320156212 + 0.05 = -0.270156212 x = -0.270156212 Simplifying x = -0.270156212

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.370156212, -0.270156212}

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