17n^2+19n=6

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Solution for 17n^2+19n=6 equation:


Simplifying
17n2 + 19n = 6

Reorder the terms:
19n + 17n2 = 6

Solving
19n + 17n2 = 6

Solving for variable 'n'.

Reorder the terms:
-6 + 19n + 17n2 = 6 + -6

Combine like terms: 6 + -6 = 0
-6 + 19n + 17n2 = 0

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

Divide each side by '17'.
-0.3529411765 + 1.117647059n + n2 = 0

Move the constant term to the right:

Add '0.3529411765' to each side of the equation.
-0.3529411765 + 1.117647059n + 0.3529411765 + n2 = 0 + 0.3529411765

Reorder the terms:
-0.3529411765 + 0.3529411765 + 1.117647059n + n2 = 0 + 0.3529411765

Combine like terms: -0.3529411765 + 0.3529411765 = 0.0000000000
0.0000000000 + 1.117647059n + n2 = 0 + 0.3529411765
1.117647059n + n2 = 0 + 0.3529411765

Combine like terms: 0 + 0.3529411765 = 0.3529411765
1.117647059n + n2 = 0.3529411765

The n term is 1.117647059n.  Take half its coefficient (0.5588235295).
Square it (0.3122837371) and add it to both sides.

Add '0.3122837371' to each side of the equation.
1.117647059n + 0.3122837371 + n2 = 0.3529411765 + 0.3122837371

Reorder the terms:
0.3122837371 + 1.117647059n + n2 = 0.3529411765 + 0.3122837371

Combine like terms: 0.3529411765 + 0.3122837371 = 0.6652249136
0.3122837371 + 1.117647059n + n2 = 0.6652249136

Factor a perfect square on the left side:
(n + 0.5588235295)(n + 0.5588235295) = 0.6652249136

Calculate the square root of the right side: 0.815613213

Break this problem into two subproblems by setting 
(n + 0.5588235295) equal to 0.815613213 and -0.815613213.

Subproblem 1

n + 0.5588235295 = 0.815613213 Simplifying n + 0.5588235295 = 0.815613213 Reorder the terms: 0.5588235295 + n = 0.815613213 Solving 0.5588235295 + n = 0.815613213 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5588235295' to each side of the equation. 0.5588235295 + -0.5588235295 + n = 0.815613213 + -0.5588235295 Combine like terms: 0.5588235295 + -0.5588235295 = 0.0000000000 0.0000000000 + n = 0.815613213 + -0.5588235295 n = 0.815613213 + -0.5588235295 Combine like terms: 0.815613213 + -0.5588235295 = 0.2567896835 n = 0.2567896835 Simplifying n = 0.2567896835

Subproblem 2

n + 0.5588235295 = -0.815613213 Simplifying n + 0.5588235295 = -0.815613213 Reorder the terms: 0.5588235295 + n = -0.815613213 Solving 0.5588235295 + n = -0.815613213 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5588235295' to each side of the equation. 0.5588235295 + -0.5588235295 + n = -0.815613213 + -0.5588235295 Combine like terms: 0.5588235295 + -0.5588235295 = 0.0000000000 0.0000000000 + n = -0.815613213 + -0.5588235295 n = -0.815613213 + -0.5588235295 Combine like terms: -0.815613213 + -0.5588235295 = -1.3744367425 n = -1.3744367425 Simplifying n = -1.3744367425

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.2567896835, -1.3744367425}

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