9n^2+7n=3

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Solution for 9n^2+7n=3 equation:


Simplifying
9n2 + 7n = 3

Reorder the terms:
7n + 9n2 = 3

Solving
7n + 9n2 = 3

Solving for variable 'n'.

Reorder the terms:
-3 + 7n + 9n2 = 3 + -3

Combine like terms: 3 + -3 = 0
-3 + 7n + 9n2 = 0

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

Divide each side by '9'.
-0.3333333333 + 0.7777777778n + n2 = 0

Move the constant term to the right:

Add '0.3333333333' to each side of the equation.
-0.3333333333 + 0.7777777778n + 0.3333333333 + n2 = 0 + 0.3333333333

Reorder the terms:
-0.3333333333 + 0.3333333333 + 0.7777777778n + n2 = 0 + 0.3333333333

Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000
0.0000000000 + 0.7777777778n + n2 = 0 + 0.3333333333
0.7777777778n + n2 = 0 + 0.3333333333

Combine like terms: 0 + 0.3333333333 = 0.3333333333
0.7777777778n + n2 = 0.3333333333

The n term is 0.7777777778n.  Take half its coefficient (0.3888888889).
Square it (0.1512345679) and add it to both sides.

Add '0.1512345679' to each side of the equation.
0.7777777778n + 0.1512345679 + n2 = 0.3333333333 + 0.1512345679

Reorder the terms:
0.1512345679 + 0.7777777778n + n2 = 0.3333333333 + 0.1512345679

Combine like terms: 0.3333333333 + 0.1512345679 = 0.4845679012
0.1512345679 + 0.7777777778n + n2 = 0.4845679012

Factor a perfect square on the left side:
(n + 0.3888888889)(n + 0.3888888889) = 0.4845679012

Calculate the square root of the right side: 0.696109116

Break this problem into two subproblems by setting 
(n + 0.3888888889) equal to 0.696109116 and -0.696109116.

Subproblem 1

n + 0.3888888889 = 0.696109116 Simplifying n + 0.3888888889 = 0.696109116 Reorder the terms: 0.3888888889 + n = 0.696109116 Solving 0.3888888889 + n = 0.696109116 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.3888888889' to each side of the equation. 0.3888888889 + -0.3888888889 + n = 0.696109116 + -0.3888888889 Combine like terms: 0.3888888889 + -0.3888888889 = 0.0000000000 0.0000000000 + n = 0.696109116 + -0.3888888889 n = 0.696109116 + -0.3888888889 Combine like terms: 0.696109116 + -0.3888888889 = 0.3072202271 n = 0.3072202271 Simplifying n = 0.3072202271

Subproblem 2

n + 0.3888888889 = -0.696109116 Simplifying n + 0.3888888889 = -0.696109116 Reorder the terms: 0.3888888889 + n = -0.696109116 Solving 0.3888888889 + n = -0.696109116 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.3888888889' to each side of the equation. 0.3888888889 + -0.3888888889 + n = -0.696109116 + -0.3888888889 Combine like terms: 0.3888888889 + -0.3888888889 = 0.0000000000 0.0000000000 + n = -0.696109116 + -0.3888888889 n = -0.696109116 + -0.3888888889 Combine like terms: -0.696109116 + -0.3888888889 = -1.0849980049 n = -1.0849980049 Simplifying n = -1.0849980049

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.3072202271, -1.0849980049}

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