5n(2+n)=3n+18

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Solution for 5n(2+n)=3n+18 equation:


Simplifying
5n(2 + n) = 3n + 18
(2 * 5n + n * 5n) = 3n + 18
(10n + 5n2) = 3n + 18

Reorder the terms:
10n + 5n2 = 18 + 3n

Solving
10n + 5n2 = 18 + 3n

Solving for variable 'n'.

Reorder the terms:
-18 + 10n + -3n + 5n2 = 18 + 3n + -18 + -3n

Combine like terms: 10n + -3n = 7n
-18 + 7n + 5n2 = 18 + 3n + -18 + -3n

Reorder the terms:
-18 + 7n + 5n2 = 18 + -18 + 3n + -3n

Combine like terms: 18 + -18 = 0
-18 + 7n + 5n2 = 0 + 3n + -3n
-18 + 7n + 5n2 = 3n + -3n

Combine like terms: 3n + -3n = 0
-18 + 7n + 5n2 = 0

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

Divide each side by '5'.
-3.6 + 1.4n + n2 = 0

Move the constant term to the right:

Add '3.6' to each side of the equation.
-3.6 + 1.4n + 3.6 + n2 = 0 + 3.6

Reorder the terms:
-3.6 + 3.6 + 1.4n + n2 = 0 + 3.6

Combine like terms: -3.6 + 3.6 = 0.0
0.0 + 1.4n + n2 = 0 + 3.6
1.4n + n2 = 0 + 3.6

Combine like terms: 0 + 3.6 = 3.6
1.4n + n2 = 3.6

The n term is 1.4n.  Take half its coefficient (0.7).
Square it (0.49) and add it to both sides.

Add '0.49' to each side of the equation.
1.4n + 0.49 + n2 = 3.6 + 0.49

Reorder the terms:
0.49 + 1.4n + n2 = 3.6 + 0.49

Combine like terms: 3.6 + 0.49 = 4.09
0.49 + 1.4n + n2 = 4.09

Factor a perfect square on the left side:
(n + 0.7)(n + 0.7) = 4.09

Calculate the square root of the right side: 2.022374842

Break this problem into two subproblems by setting 
(n + 0.7) equal to 2.022374842 and -2.022374842.

Subproblem 1

n + 0.7 = 2.022374842 Simplifying n + 0.7 = 2.022374842 Reorder the terms: 0.7 + n = 2.022374842 Solving 0.7 + n = 2.022374842 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.7' to each side of the equation. 0.7 + -0.7 + n = 2.022374842 + -0.7 Combine like terms: 0.7 + -0.7 = 0.0 0.0 + n = 2.022374842 + -0.7 n = 2.022374842 + -0.7 Combine like terms: 2.022374842 + -0.7 = 1.322374842 n = 1.322374842 Simplifying n = 1.322374842

Subproblem 2

n + 0.7 = -2.022374842 Simplifying n + 0.7 = -2.022374842 Reorder the terms: 0.7 + n = -2.022374842 Solving 0.7 + n = -2.022374842 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.7' to each side of the equation. 0.7 + -0.7 + n = -2.022374842 + -0.7 Combine like terms: 0.7 + -0.7 = 0.0 0.0 + n = -2.022374842 + -0.7 n = -2.022374842 + -0.7 Combine like terms: -2.022374842 + -0.7 = -2.722374842 n = -2.722374842 Simplifying n = -2.722374842

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.322374842, -2.722374842}

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