3n(5n+9)=11

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Solution for 3n(5n+9)=11 equation:


Simplifying
3n(5n + 9) = 11

Reorder the terms:
3n(9 + 5n) = 11
(9 * 3n + 5n * 3n) = 11
(27n + 15n2) = 11

Solving
27n + 15n2 = 11

Solving for variable 'n'.

Reorder the terms:
-11 + 27n + 15n2 = 11 + -11

Combine like terms: 11 + -11 = 0
-11 + 27n + 15n2 = 0

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

Divide each side by '15'.
-0.7333333333 + 1.8n + n2 = 0

Move the constant term to the right:

Add '0.7333333333' to each side of the equation.
-0.7333333333 + 1.8n + 0.7333333333 + n2 = 0 + 0.7333333333

Reorder the terms:
-0.7333333333 + 0.7333333333 + 1.8n + n2 = 0 + 0.7333333333

Combine like terms: -0.7333333333 + 0.7333333333 = 0.0000000000
0.0000000000 + 1.8n + n2 = 0 + 0.7333333333
1.8n + n2 = 0 + 0.7333333333

Combine like terms: 0 + 0.7333333333 = 0.7333333333
1.8n + n2 = 0.7333333333

The n term is 1.8n.  Take half its coefficient (0.9).
Square it (0.81) and add it to both sides.

Add '0.81' to each side of the equation.
1.8n + 0.81 + n2 = 0.7333333333 + 0.81

Reorder the terms:
0.81 + 1.8n + n2 = 0.7333333333 + 0.81

Combine like terms: 0.7333333333 + 0.81 = 1.5433333333
0.81 + 1.8n + n2 = 1.5433333333

Factor a perfect square on the left side:
(n + 0.9)(n + 0.9) = 1.5433333333

Calculate the square root of the right side: 1.242309677

Break this problem into two subproblems by setting 
(n + 0.9) equal to 1.242309677 and -1.242309677.

Subproblem 1

n + 0.9 = 1.242309677 Simplifying n + 0.9 = 1.242309677 Reorder the terms: 0.9 + n = 1.242309677 Solving 0.9 + n = 1.242309677 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.9' to each side of the equation. 0.9 + -0.9 + n = 1.242309677 + -0.9 Combine like terms: 0.9 + -0.9 = 0.0 0.0 + n = 1.242309677 + -0.9 n = 1.242309677 + -0.9 Combine like terms: 1.242309677 + -0.9 = 0.342309677 n = 0.342309677 Simplifying n = 0.342309677

Subproblem 2

n + 0.9 = -1.242309677 Simplifying n + 0.9 = -1.242309677 Reorder the terms: 0.9 + n = -1.242309677 Solving 0.9 + n = -1.242309677 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.9' to each side of the equation. 0.9 + -0.9 + n = -1.242309677 + -0.9 Combine like terms: 0.9 + -0.9 = 0.0 0.0 + n = -1.242309677 + -0.9 n = -1.242309677 + -0.9 Combine like terms: -1.242309677 + -0.9 = -2.142309677 n = -2.142309677 Simplifying n = -2.142309677

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.342309677, -2.142309677}

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