2(n+n^2)=45

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Solution for 2(n+n^2)=45 equation:


Simplifying
2(n + n2) = 45
(n * 2 + n2 * 2) = 45
(2n + 2n2) = 45

Solving
2n + 2n2 = 45

Solving for variable 'n'.

Reorder the terms:
-45 + 2n + 2n2 = 45 + -45

Combine like terms: 45 + -45 = 0
-45 + 2n + 2n2 = 0

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

Divide each side by '2'.
-22.5 + n + n2 = 0

Move the constant term to the right:

Add '22.5' to each side of the equation.
-22.5 + n + 22.5 + n2 = 0 + 22.5

Reorder the terms:
-22.5 + 22.5 + n + n2 = 0 + 22.5

Combine like terms: -22.5 + 22.5 = 0.0
0.0 + n + n2 = 0 + 22.5
n + n2 = 0 + 22.5

Combine like terms: 0 + 22.5 = 22.5
n + n2 = 22.5

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.
 + 0.25 + n2 = 22.5 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + n2 = 22.5 + 0.25

Combine like terms: 22.5 + 0.25 = 22.75
1.25 + n2 = 22.75

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 22.75

Calculate the square root of the right side: 4.769696007

Break this problem into two subproblems by setting 
(n + 0.5) equal to 4.769696007 and -4.769696007.

Subproblem 1

n + 0.5 = 4.769696007 Simplifying n + 0.5 = 4.769696007 Reorder the terms: 0.5 + n = 4.769696007 Solving 0.5 + n = 4.769696007 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 = 4.769696007 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 4.769696007 + -0.5 n = 4.769696007 + -0.5 Combine like terms: 4.769696007 + -0.5 = 4.269696007 n = 4.269696007 Simplifying n = 4.269696007

Subproblem 2

n + 0.5 = -4.769696007 Simplifying n + 0.5 = -4.769696007 Reorder the terms: 0.5 + n = -4.769696007 Solving 0.5 + n = -4.769696007 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 = -4.769696007 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -4.769696007 + -0.5 n = -4.769696007 + -0.5 Combine like terms: -4.769696007 + -0.5 = -5.269696007 n = -5.269696007 Simplifying n = -5.269696007

Solution

The solution to the problem is based on the solutions from the subproblems. n = {4.269696007, -5.269696007}

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