n^2-8n-19=0

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


Simplifying
n2 + -8n + -19 = 0

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

Solving
-19 + -8n + n2 = 0

Solving for variable 'n'.

Begin completing the square.

Move the constant term to the right:

Add '19' to each side of the equation.
-19 + -8n + 19 + n2 = 0 + 19

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

Combine like terms: -19 + 19 = 0
0 + -8n + n2 = 0 + 19
-8n + n2 = 0 + 19

Combine like terms: 0 + 19 = 19
-8n + n2 = 19

The n term is -8n.  Take half its coefficient (-4).
Square it (16) and add it to both sides.

Add '16' to each side of the equation.
-8n + 16 + n2 = 19 + 16

Reorder the terms:
16 + -8n + n2 = 19 + 16

Combine like terms: 19 + 16 = 35
16 + -8n + n2 = 35

Factor a perfect square on the left side:
(n + -4)(n + -4) = 35

Calculate the square root of the right side: 5.916079783

Break this problem into two subproblems by setting 
(n + -4) equal to 5.916079783 and -5.916079783.

Subproblem 1

n + -4 = 5.916079783 Simplifying n + -4 = 5.916079783 Reorder the terms: -4 + n = 5.916079783 Solving -4 + n = 5.916079783 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + n = 5.916079783 + 4 Combine like terms: -4 + 4 = 0 0 + n = 5.916079783 + 4 n = 5.916079783 + 4 Combine like terms: 5.916079783 + 4 = 9.916079783 n = 9.916079783 Simplifying n = 9.916079783

Subproblem 2

n + -4 = -5.916079783 Simplifying n + -4 = -5.916079783 Reorder the terms: -4 + n = -5.916079783 Solving -4 + n = -5.916079783 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + n = -5.916079783 + 4 Combine like terms: -4 + 4 = 0 0 + n = -5.916079783 + 4 n = -5.916079783 + 4 Combine like terms: -5.916079783 + 4 = -1.916079783 n = -1.916079783 Simplifying n = -1.916079783

Solution

The solution to the problem is based on the solutions from the subproblems. n = {9.916079783, -1.916079783}

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