5n^2+19n-69=-2

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Solution for 5n^2+19n-69=-2 equation:


Simplifying
5n2 + 19n + -69 = -2

Reorder the terms:
-69 + 19n + 5n2 = -2

Solving
-69 + 19n + 5n2 = -2

Solving for variable 'n'.

Reorder the terms:
-69 + 2 + 19n + 5n2 = -2 + 2

Combine like terms: -69 + 2 = -67
-67 + 19n + 5n2 = -2 + 2

Combine like terms: -2 + 2 = 0
-67 + 19n + 5n2 = 0

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

Divide each side by '5'.
-13.4 + 3.8n + n2 = 0

Move the constant term to the right:

Add '13.4' to each side of the equation.
-13.4 + 3.8n + 13.4 + n2 = 0 + 13.4

Reorder the terms:
-13.4 + 13.4 + 3.8n + n2 = 0 + 13.4

Combine like terms: -13.4 + 13.4 = 0.0
0.0 + 3.8n + n2 = 0 + 13.4
3.8n + n2 = 0 + 13.4

Combine like terms: 0 + 13.4 = 13.4
3.8n + n2 = 13.4

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 = 13.4 + 3.61

Reorder the terms:
3.61 + 3.8n + n2 = 13.4 + 3.61

Combine like terms: 13.4 + 3.61 = 17.01
3.61 + 3.8n + n2 = 17.01

Factor a perfect square on the left side:
(n + 1.9)(n + 1.9) = 17.01

Calculate the square root of the right side: 4.124318125

Break this problem into two subproblems by setting 
(n + 1.9) equal to 4.124318125 and -4.124318125.

Subproblem 1

n + 1.9 = 4.124318125 Simplifying n + 1.9 = 4.124318125 Reorder the terms: 1.9 + n = 4.124318125 Solving 1.9 + n = 4.124318125 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 = 4.124318125 + -1.9 Combine like terms: 1.9 + -1.9 = 0.0 0.0 + n = 4.124318125 + -1.9 n = 4.124318125 + -1.9 Combine like terms: 4.124318125 + -1.9 = 2.224318125 n = 2.224318125 Simplifying n = 2.224318125

Subproblem 2

n + 1.9 = -4.124318125 Simplifying n + 1.9 = -4.124318125 Reorder the terms: 1.9 + n = -4.124318125 Solving 1.9 + n = -4.124318125 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 = -4.124318125 + -1.9 Combine like terms: 1.9 + -1.9 = 0.0 0.0 + n = -4.124318125 + -1.9 n = -4.124318125 + -1.9 Combine like terms: -4.124318125 + -1.9 = -6.024318125 n = -6.024318125 Simplifying n = -6.024318125

Solution

The solution to the problem is based on the solutions from the subproblems. n = {2.224318125, -6.024318125}

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