5n^2+19n=-2

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Solution for 5n^2+19n=-2 equation:


Simplifying
5n2 + 19n = -2

Reorder the terms:
19n + 5n2 = -2

Solving
19n + 5n2 = -2

Solving for variable 'n'.

Reorder the terms:
2 + 19n + 5n2 = -2 + 2

Combine like terms: -2 + 2 = 0
2 + 19n + 5n2 = 0

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

Divide each side by '5'.
0.4 + 3.8n + n2 = 0

Move the constant term to the right:

Add '-0.4' to each side of the equation.
0.4 + 3.8n + -0.4 + n2 = 0 + -0.4

Reorder the terms:
0.4 + -0.4 + 3.8n + n2 = 0 + -0.4

Combine like terms: 0.4 + -0.4 = 0.0
0.0 + 3.8n + n2 = 0 + -0.4
3.8n + n2 = 0 + -0.4

Combine like terms: 0 + -0.4 = -0.4
3.8n + n2 = -0.4

The n term is 3.8n.  Take half its coefficient (1.9).
Square it (3.61) and add it to both sides.

Add '3.61' to each side of the equation.
3.8n + 3.61 + n2 = -0.4 + 3.61

Reorder the terms:
3.61 + 3.8n + n2 = -0.4 + 3.61

Combine like terms: -0.4 + 3.61 = 3.21
3.61 + 3.8n + n2 = 3.21

Factor a perfect square on the left side:
(n + 1.9)(n + 1.9) = 3.21

Calculate the square root of the right side: 1.791647287

Break this problem into two subproblems by setting 
(n + 1.9) equal to 1.791647287 and -1.791647287.

Subproblem 1

n + 1.9 = 1.791647287 Simplifying n + 1.9 = 1.791647287 Reorder the terms: 1.9 + n = 1.791647287 Solving 1.9 + n = 1.791647287 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.9' to each side of the equation. 1.9 + -1.9 + n = 1.791647287 + -1.9 Combine like terms: 1.9 + -1.9 = 0.0 0.0 + n = 1.791647287 + -1.9 n = 1.791647287 + -1.9 Combine like terms: 1.791647287 + -1.9 = -0.108352713 n = -0.108352713 Simplifying n = -0.108352713

Subproblem 2

n + 1.9 = -1.791647287 Simplifying n + 1.9 = -1.791647287 Reorder the terms: 1.9 + n = -1.791647287 Solving 1.9 + n = -1.791647287 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.9' to each side of the equation. 1.9 + -1.9 + n = -1.791647287 + -1.9 Combine like terms: 1.9 + -1.9 = 0.0 0.0 + n = -1.791647287 + -1.9 n = -1.791647287 + -1.9 Combine like terms: -1.791647287 + -1.9 = -3.691647287 n = -3.691647287 Simplifying n = -3.691647287

Solution

The solution to the problem is based on the solutions from the subproblems. n = {-0.108352713, -3.691647287}

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