5n^2+19n-68=2

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Solution for 5n^2+19n-68=2 equation:


Simplifying
5n2 + 19n + -68 = 2

Reorder the terms:
-68 + 19n + 5n2 = 2

Solving
-68 + 19n + 5n2 = 2

Solving for variable 'n'.

Reorder the terms:
-68 + -2 + 19n + 5n2 = 2 + -2

Combine like terms: -68 + -2 = -70
-70 + 19n + 5n2 = 2 + -2

Combine like terms: 2 + -2 = 0
-70 + 19n + 5n2 = 0

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

Divide each side by '5'.
-14 + 3.8n + n2 = 0

Move the constant term to the right:

Add '14' to each side of the equation.
-14 + 3.8n + 14 + n2 = 0 + 14

Reorder the terms:
-14 + 14 + 3.8n + n2 = 0 + 14

Combine like terms: -14 + 14 = 0
0 + 3.8n + n2 = 0 + 14
3.8n + n2 = 0 + 14

Combine like terms: 0 + 14 = 14
3.8n + n2 = 14

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 = 14 + 3.61

Reorder the terms:
3.61 + 3.8n + n2 = 14 + 3.61

Combine like terms: 14 + 3.61 = 17.61
3.61 + 3.8n + n2 = 17.61

Factor a perfect square on the left side:
(n + 1.9)(n + 1.9) = 17.61

Calculate the square root of the right side: 4.196427052

Break this problem into two subproblems by setting 
(n + 1.9) equal to 4.196427052 and -4.196427052.

Subproblem 1

n + 1.9 = 4.196427052 Simplifying n + 1.9 = 4.196427052 Reorder the terms: 1.9 + n = 4.196427052 Solving 1.9 + n = 4.196427052 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 = 4.196427052 + -1.9 Combine like terms: 1.9 + -1.9 = 0.0 0.0 + n = 4.196427052 + -1.9 n = 4.196427052 + -1.9 Combine like terms: 4.196427052 + -1.9 = 2.296427052 n = 2.296427052 Simplifying n = 2.296427052

Subproblem 2

n + 1.9 = -4.196427052 Simplifying n + 1.9 = -4.196427052 Reorder the terms: 1.9 + n = -4.196427052 Solving 1.9 + n = -4.196427052 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 = -4.196427052 + -1.9 Combine like terms: 1.9 + -1.9 = 0.0 0.0 + n = -4.196427052 + -1.9 n = -4.196427052 + -1.9 Combine like terms: -4.196427052 + -1.9 = -6.096427052 n = -6.096427052 Simplifying n = -6.096427052

Solution

The solution to the problem is based on the solutions from the subproblems. n = {2.296427052, -6.096427052}

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