5n^2+42n+13=0

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Solution for 5n^2+42n+13=0 equation:


Simplifying
5n2 + 42n + 13 = 0

Reorder the terms:
13 + 42n + 5n2 = 0

Solving
13 + 42n + 5n2 = 0

Solving for variable 'n'.

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

Divide each side by '5'.
2.6 + 8.4n + n2 = 0

Move the constant term to the right:

Add '-2.6' to each side of the equation.
2.6 + 8.4n + -2.6 + n2 = 0 + -2.6

Reorder the terms:
2.6 + -2.6 + 8.4n + n2 = 0 + -2.6

Combine like terms: 2.6 + -2.6 = 0.0
0.0 + 8.4n + n2 = 0 + -2.6
8.4n + n2 = 0 + -2.6

Combine like terms: 0 + -2.6 = -2.6
8.4n + n2 = -2.6

The n term is 8.4n.  Take half its coefficient (4.2).
Square it (17.64) and add it to both sides.

Add '17.64' to each side of the equation.
8.4n + 17.64 + n2 = -2.6 + 17.64

Reorder the terms:
17.64 + 8.4n + n2 = -2.6 + 17.64

Combine like terms: -2.6 + 17.64 = 15.04
17.64 + 8.4n + n2 = 15.04

Factor a perfect square on the left side:
(n + 4.2)(n + 4.2) = 15.04

Calculate the square root of the right side: 3.878143886

Break this problem into two subproblems by setting 
(n + 4.2) equal to 3.878143886 and -3.878143886.

Subproblem 1

n + 4.2 = 3.878143886 Simplifying n + 4.2 = 3.878143886 Reorder the terms: 4.2 + n = 3.878143886 Solving 4.2 + n = 3.878143886 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-4.2' to each side of the equation. 4.2 + -4.2 + n = 3.878143886 + -4.2 Combine like terms: 4.2 + -4.2 = 0.0 0.0 + n = 3.878143886 + -4.2 n = 3.878143886 + -4.2 Combine like terms: 3.878143886 + -4.2 = -0.321856114 n = -0.321856114 Simplifying n = -0.321856114

Subproblem 2

n + 4.2 = -3.878143886 Simplifying n + 4.2 = -3.878143886 Reorder the terms: 4.2 + n = -3.878143886 Solving 4.2 + n = -3.878143886 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-4.2' to each side of the equation. 4.2 + -4.2 + n = -3.878143886 + -4.2 Combine like terms: 4.2 + -4.2 = 0.0 0.0 + n = -3.878143886 + -4.2 n = -3.878143886 + -4.2 Combine like terms: -3.878143886 + -4.2 = -8.078143886 n = -8.078143886 Simplifying n = -8.078143886

Solution

The solution to the problem is based on the solutions from the subproblems. n = {-0.321856114, -8.078143886}

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