-3n^2+34n=8n^2+12

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Solution for -3n^2+34n=8n^2+12 equation:


Simplifying
-3n2 + 34n = 8n2 + 12

Reorder the terms:
34n + -3n2 = 8n2 + 12

Reorder the terms:
34n + -3n2 = 12 + 8n2

Solving
34n + -3n2 = 12 + 8n2

Solving for variable 'n'.

Reorder the terms:
-12 + 34n + -3n2 + -8n2 = 12 + 8n2 + -12 + -8n2

Combine like terms: -3n2 + -8n2 = -11n2
-12 + 34n + -11n2 = 12 + 8n2 + -12 + -8n2

Reorder the terms:
-12 + 34n + -11n2 = 12 + -12 + 8n2 + -8n2

Combine like terms: 12 + -12 = 0
-12 + 34n + -11n2 = 0 + 8n2 + -8n2
-12 + 34n + -11n2 = 8n2 + -8n2

Combine like terms: 8n2 + -8n2 = 0
-12 + 34n + -11n2 = 0

Begin completing the square.  Divide all terms by
-11 the coefficient of the squared term: 

Divide each side by '-11'.
1.090909091 + -3.090909091n + n2 = 0

Move the constant term to the right:

Add '-1.090909091' to each side of the equation.
1.090909091 + -3.090909091n + -1.090909091 + n2 = 0 + -1.090909091

Reorder the terms:
1.090909091 + -1.090909091 + -3.090909091n + n2 = 0 + -1.090909091

Combine like terms: 1.090909091 + -1.090909091 = 0.000000000
0.000000000 + -3.090909091n + n2 = 0 + -1.090909091
-3.090909091n + n2 = 0 + -1.090909091

Combine like terms: 0 + -1.090909091 = -1.090909091
-3.090909091n + n2 = -1.090909091

The n term is -3.090909091n.  Take half its coefficient (-1.545454546).
Square it (2.388429754) and add it to both sides.

Add '2.388429754' to each side of the equation.
-3.090909091n + 2.388429754 + n2 = -1.090909091 + 2.388429754

Reorder the terms:
2.388429754 + -3.090909091n + n2 = -1.090909091 + 2.388429754

Combine like terms: -1.090909091 + 2.388429754 = 1.297520663
2.388429754 + -3.090909091n + n2 = 1.297520663

Factor a perfect square on the left side:
(n + -1.545454546)(n + -1.545454546) = 1.297520663

Calculate the square root of the right side: 1.139087645

Break this problem into two subproblems by setting 
(n + -1.545454546) equal to 1.139087645 and -1.139087645.

Subproblem 1

n + -1.545454546 = 1.139087645 Simplifying n + -1.545454546 = 1.139087645 Reorder the terms: -1.545454546 + n = 1.139087645 Solving -1.545454546 + n = 1.139087645 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.545454546' to each side of the equation. -1.545454546 + 1.545454546 + n = 1.139087645 + 1.545454546 Combine like terms: -1.545454546 + 1.545454546 = 0.000000000 0.000000000 + n = 1.139087645 + 1.545454546 n = 1.139087645 + 1.545454546 Combine like terms: 1.139087645 + 1.545454546 = 2.684542191 n = 2.684542191 Simplifying n = 2.684542191

Subproblem 2

n + -1.545454546 = -1.139087645 Simplifying n + -1.545454546 = -1.139087645 Reorder the terms: -1.545454546 + n = -1.139087645 Solving -1.545454546 + n = -1.139087645 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.545454546' to each side of the equation. -1.545454546 + 1.545454546 + n = -1.139087645 + 1.545454546 Combine like terms: -1.545454546 + 1.545454546 = 0.000000000 0.000000000 + n = -1.139087645 + 1.545454546 n = -1.139087645 + 1.545454546 Combine like terms: -1.139087645 + 1.545454546 = 0.406366901 n = 0.406366901 Simplifying n = 0.406366901

Solution

The solution to the problem is based on the solutions from the subproblems. n = {2.684542191, 0.406366901}

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