2n^2-8n-19=0

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Solution for 2n^2-8n-19=0 equation:


Simplifying
2n2 + -8n + -19 = 0

Reorder the terms:
-19 + -8n + 2n2 = 0

Solving
-19 + -8n + 2n2 = 0

Solving for variable 'n'.

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

Divide each side by '2'.
-9.5 + -4n + n2 = 0

Move the constant term to the right:

Add '9.5' to each side of the equation.
-9.5 + -4n + 9.5 + n2 = 0 + 9.5

Reorder the terms:
-9.5 + 9.5 + -4n + n2 = 0 + 9.5

Combine like terms: -9.5 + 9.5 = 0.0
0.0 + -4n + n2 = 0 + 9.5
-4n + n2 = 0 + 9.5

Combine like terms: 0 + 9.5 = 9.5
-4n + n2 = 9.5

The n term is -4n.  Take half its coefficient (-2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
-4n + 4 + n2 = 9.5 + 4

Reorder the terms:
4 + -4n + n2 = 9.5 + 4

Combine like terms: 9.5 + 4 = 13.5
4 + -4n + n2 = 13.5

Factor a perfect square on the left side:
(n + -2)(n + -2) = 13.5

Calculate the square root of the right side: 3.674234614

Break this problem into two subproblems by setting 
(n + -2) equal to 3.674234614 and -3.674234614.

Subproblem 1

n + -2 = 3.674234614 Simplifying n + -2 = 3.674234614 Reorder the terms: -2 + n = 3.674234614 Solving -2 + n = 3.674234614 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + n = 3.674234614 + 2 Combine like terms: -2 + 2 = 0 0 + n = 3.674234614 + 2 n = 3.674234614 + 2 Combine like terms: 3.674234614 + 2 = 5.674234614 n = 5.674234614 Simplifying n = 5.674234614

Subproblem 2

n + -2 = -3.674234614 Simplifying n + -2 = -3.674234614 Reorder the terms: -2 + n = -3.674234614 Solving -2 + n = -3.674234614 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + n = -3.674234614 + 2 Combine like terms: -2 + 2 = 0 0 + n = -3.674234614 + 2 n = -3.674234614 + 2 Combine like terms: -3.674234614 + 2 = -1.674234614 n = -1.674234614 Simplifying n = -1.674234614

Solution

The solution to the problem is based on the solutions from the subproblems. n = {5.674234614, -1.674234614}

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