10y^2+41y=-9

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Solution for 10y^2+41y=-9 equation:


Simplifying
10y2 + 41y = -9

Reorder the terms:
41y + 10y2 = -9

Solving
41y + 10y2 = -9

Solving for variable 'y'.

Reorder the terms:
9 + 41y + 10y2 = -9 + 9

Combine like terms: -9 + 9 = 0
9 + 41y + 10y2 = 0

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

Divide each side by '10'.
0.9 + 4.1y + y2 = 0

Move the constant term to the right:

Add '-0.9' to each side of the equation.
0.9 + 4.1y + -0.9 + y2 = 0 + -0.9

Reorder the terms:
0.9 + -0.9 + 4.1y + y2 = 0 + -0.9

Combine like terms: 0.9 + -0.9 = 0.0
0.0 + 4.1y + y2 = 0 + -0.9
4.1y + y2 = 0 + -0.9

Combine like terms: 0 + -0.9 = -0.9
4.1y + y2 = -0.9

The y term is 4.1y.  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.
4.1y + 4.2025 + y2 = -0.9 + 4.2025

Reorder the terms:
4.2025 + 4.1y + y2 = -0.9 + 4.2025

Combine like terms: -0.9 + 4.2025 = 3.3025
4.2025 + 4.1y + y2 = 3.3025

Factor a perfect square on the left side:
(y + 2.05)(y + 2.05) = 3.3025

Calculate the square root of the right side: 1.817278185

Break this problem into two subproblems by setting 
(y + 2.05) equal to 1.817278185 and -1.817278185.

Subproblem 1

y + 2.05 = 1.817278185 Simplifying y + 2.05 = 1.817278185 Reorder the terms: 2.05 + y = 1.817278185 Solving 2.05 + y = 1.817278185 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-2.05' to each side of the equation. 2.05 + -2.05 + y = 1.817278185 + -2.05 Combine like terms: 2.05 + -2.05 = 0.00 0.00 + y = 1.817278185 + -2.05 y = 1.817278185 + -2.05 Combine like terms: 1.817278185 + -2.05 = -0.232721815 y = -0.232721815 Simplifying y = -0.232721815

Subproblem 2

y + 2.05 = -1.817278185 Simplifying y + 2.05 = -1.817278185 Reorder the terms: 2.05 + y = -1.817278185 Solving 2.05 + y = -1.817278185 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-2.05' to each side of the equation. 2.05 + -2.05 + y = -1.817278185 + -2.05 Combine like terms: 2.05 + -2.05 = 0.00 0.00 + y = -1.817278185 + -2.05 y = -1.817278185 + -2.05 Combine like terms: -1.817278185 + -2.05 = -3.867278185 y = -3.867278185 Simplifying y = -3.867278185

Solution

The solution to the problem is based on the solutions from the subproblems. y = {-0.232721815, -3.867278185}

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