9n^2+n=4

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Solution for 9n^2+n=4 equation:


Simplifying
9n2 + n = 4

Reorder the terms:
n + 9n2 = 4

Solving
n + 9n2 = 4

Solving for variable 'n'.

Reorder the terms:
-4 + n + 9n2 = 4 + -4

Combine like terms: 4 + -4 = 0
-4 + n + 9n2 = 0

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

Divide each side by '9'.
-0.4444444444 + 0.1111111111n + n2 = 0

Move the constant term to the right:

Add '0.4444444444' to each side of the equation.
-0.4444444444 + 0.1111111111n + 0.4444444444 + n2 = 0 + 0.4444444444

Reorder the terms:
-0.4444444444 + 0.4444444444 + 0.1111111111n + n2 = 0 + 0.4444444444

Combine like terms: -0.4444444444 + 0.4444444444 = 0.0000000000
0.0000000000 + 0.1111111111n + n2 = 0 + 0.4444444444
0.1111111111n + n2 = 0 + 0.4444444444

Combine like terms: 0 + 0.4444444444 = 0.4444444444
0.1111111111n + n2 = 0.4444444444

The n term is n.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
0.1111111111n + 0.25 + n2 = 0.4444444444 + 0.25

Reorder the terms:
0.25 + 0.1111111111n + n2 = 0.4444444444 + 0.25

Combine like terms: 0.4444444444 + 0.25 = 0.6944444444
0.25 + 0.1111111111n + n2 = 0.6944444444

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 0.6944444444

Calculate the square root of the right side: 0.833333333

Break this problem into two subproblems by setting 
(n + 0.5) equal to 0.833333333 and -0.833333333.

Subproblem 1

n + 0.5 = 0.833333333 Simplifying n + 0.5 = 0.833333333 Reorder the terms: 0.5 + n = 0.833333333 Solving 0.5 + n = 0.833333333 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = 0.833333333 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 0.833333333 + -0.5 n = 0.833333333 + -0.5 Combine like terms: 0.833333333 + -0.5 = 0.333333333 n = 0.333333333 Simplifying n = 0.333333333

Subproblem 2

n + 0.5 = -0.833333333 Simplifying n + 0.5 = -0.833333333 Reorder the terms: 0.5 + n = -0.833333333 Solving 0.5 + n = -0.833333333 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = -0.833333333 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -0.833333333 + -0.5 n = -0.833333333 + -0.5 Combine like terms: -0.833333333 + -0.5 = -1.333333333 n = -1.333333333 Simplifying n = -1.333333333

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.333333333, -1.333333333}

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