10x^2+41x+6=0

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Solution for 10x^2+41x+6=0 equation:


Simplifying
10x2 + 41x + 6 = 0

Reorder the terms:
6 + 41x + 10x2 = 0

Solving
6 + 41x + 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'.
0.6 + 4.1x + x2 = 0

Move the constant term to the right:

Add '-0.6' to each side of the equation.
0.6 + 4.1x + -0.6 + x2 = 0 + -0.6

Reorder the terms:
0.6 + -0.6 + 4.1x + x2 = 0 + -0.6

Combine like terms: 0.6 + -0.6 = 0.0
0.0 + 4.1x + x2 = 0 + -0.6
4.1x + x2 = 0 + -0.6

Combine like terms: 0 + -0.6 = -0.6
4.1x + x2 = -0.6

The x term is 4.1x.  Take half its coefficient (2.05).
Square it (4.2025) and add it to both sides.

Add '4.2025' to each side of the equation.
4.1x + 4.2025 + x2 = -0.6 + 4.2025

Reorder the terms:
4.2025 + 4.1x + x2 = -0.6 + 4.2025

Combine like terms: -0.6 + 4.2025 = 3.6025
4.2025 + 4.1x + x2 = 3.6025

Factor a perfect square on the left side:
(x + 2.05)(x + 2.05) = 3.6025

Calculate the square root of the right side: 1.89802529

Break this problem into two subproblems by setting 
(x + 2.05) equal to 1.89802529 and -1.89802529.

Subproblem 1

x + 2.05 = 1.89802529 Simplifying x + 2.05 = 1.89802529 Reorder the terms: 2.05 + x = 1.89802529 Solving 2.05 + x = 1.89802529 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.05' to each side of the equation. 2.05 + -2.05 + x = 1.89802529 + -2.05 Combine like terms: 2.05 + -2.05 = 0.00 0.00 + x = 1.89802529 + -2.05 x = 1.89802529 + -2.05 Combine like terms: 1.89802529 + -2.05 = -0.15197471 x = -0.15197471 Simplifying x = -0.15197471

Subproblem 2

x + 2.05 = -1.89802529 Simplifying x + 2.05 = -1.89802529 Reorder the terms: 2.05 + x = -1.89802529 Solving 2.05 + x = -1.89802529 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.05' to each side of the equation. 2.05 + -2.05 + x = -1.89802529 + -2.05 Combine like terms: 2.05 + -2.05 = 0.00 0.00 + x = -1.89802529 + -2.05 x = -1.89802529 + -2.05 Combine like terms: -1.89802529 + -2.05 = -3.94802529 x = -3.94802529 Simplifying x = -3.94802529

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.15197471, -3.94802529}

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