n^2+19n=6

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Solution for n^2+19n=6 equation:


Simplifying
n2 + 19n = 6

Reorder the terms:
19n + n2 = 6

Solving
19n + n2 = 6

Solving for variable 'n'.

Reorder the terms:
-6 + 19n + n2 = 6 + -6

Combine like terms: 6 + -6 = 0
-6 + 19n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '6' to each side of the equation.
-6 + 19n + 6 + n2 = 0 + 6

Reorder the terms:
-6 + 6 + 19n + n2 = 0 + 6

Combine like terms: -6 + 6 = 0
0 + 19n + n2 = 0 + 6
19n + n2 = 0 + 6

Combine like terms: 0 + 6 = 6
19n + n2 = 6

The n term is 19n.  Take half its coefficient (9.5).
Square it (90.25) and add it to both sides.

Add '90.25' to each side of the equation.
19n + 90.25 + n2 = 6 + 90.25

Reorder the terms:
90.25 + 19n + n2 = 6 + 90.25

Combine like terms: 6 + 90.25 = 96.25
90.25 + 19n + n2 = 96.25

Factor a perfect square on the left side:
(n + 9.5)(n + 9.5) = 96.25

Calculate the square root of the right side: 9.810708435

Break this problem into two subproblems by setting 
(n + 9.5) equal to 9.810708435 and -9.810708435.

Subproblem 1

n + 9.5 = 9.810708435 Simplifying n + 9.5 = 9.810708435 Reorder the terms: 9.5 + n = 9.810708435 Solving 9.5 + n = 9.810708435 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-9.5' to each side of the equation. 9.5 + -9.5 + n = 9.810708435 + -9.5 Combine like terms: 9.5 + -9.5 = 0.0 0.0 + n = 9.810708435 + -9.5 n = 9.810708435 + -9.5 Combine like terms: 9.810708435 + -9.5 = 0.310708435 n = 0.310708435 Simplifying n = 0.310708435

Subproblem 2

n + 9.5 = -9.810708435 Simplifying n + 9.5 = -9.810708435 Reorder the terms: 9.5 + n = -9.810708435 Solving 9.5 + n = -9.810708435 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-9.5' to each side of the equation. 9.5 + -9.5 + n = -9.810708435 + -9.5 Combine like terms: 9.5 + -9.5 = 0.0 0.0 + n = -9.810708435 + -9.5 n = -9.810708435 + -9.5 Combine like terms: -9.810708435 + -9.5 = -19.310708435 n = -19.310708435 Simplifying n = -19.310708435

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.310708435, -19.310708435}

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