10x^2+5x-1=0

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Solution for 10x^2+5x-1=0 equation:


Simplifying
10x2 + 5x + -1 = 0

Reorder the terms:
-1 + 5x + 10x2 = 0

Solving
-1 + 5x + 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 + 0.5x + x2 = 0

Move the constant term to the right:

Add '0.1' to each side of the equation.
-0.1 + 0.5x + 0.1 + x2 = 0 + 0.1

Reorder the terms:
-0.1 + 0.1 + 0.5x + x2 = 0 + 0.1

Combine like terms: -0.1 + 0.1 = 0.0
0.0 + 0.5x + x2 = 0 + 0.1
0.5x + x2 = 0 + 0.1

Combine like terms: 0 + 0.1 = 0.1
0.5x + x2 = 0.1

The x term is 0.5x.  Take half its coefficient (0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
0.5x + 0.0625 + x2 = 0.1 + 0.0625

Reorder the terms:
0.0625 + 0.5x + x2 = 0.1 + 0.0625

Combine like terms: 0.1 + 0.0625 = 0.1625
0.0625 + 0.5x + x2 = 0.1625

Factor a perfect square on the left side:
(x + 0.25)(x + 0.25) = 0.1625

Calculate the square root of the right side: 0.403112887

Break this problem into two subproblems by setting 
(x + 0.25) equal to 0.403112887 and -0.403112887.

Subproblem 1

x + 0.25 = 0.403112887 Simplifying x + 0.25 = 0.403112887 Reorder the terms: 0.25 + x = 0.403112887 Solving 0.25 + x = 0.403112887 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = 0.403112887 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = 0.403112887 + -0.25 x = 0.403112887 + -0.25 Combine like terms: 0.403112887 + -0.25 = 0.153112887 x = 0.153112887 Simplifying x = 0.153112887

Subproblem 2

x + 0.25 = -0.403112887 Simplifying x + 0.25 = -0.403112887 Reorder the terms: 0.25 + x = -0.403112887 Solving 0.25 + x = -0.403112887 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = -0.403112887 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = -0.403112887 + -0.25 x = -0.403112887 + -0.25 Combine like terms: -0.403112887 + -0.25 = -0.653112887 x = -0.653112887 Simplifying x = -0.653112887

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.153112887, -0.653112887}

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