3n^2+12n-90=-8

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Solution for 3n^2+12n-90=-8 equation:


Simplifying
3n2 + 12n + -90 = -8

Reorder the terms:
-90 + 12n + 3n2 = -8

Solving
-90 + 12n + 3n2 = -8

Solving for variable 'n'.

Reorder the terms:
-90 + 8 + 12n + 3n2 = -8 + 8

Combine like terms: -90 + 8 = -82
-82 + 12n + 3n2 = -8 + 8

Combine like terms: -8 + 8 = 0
-82 + 12n + 3n2 = 0

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

Divide each side by '3'.
-27.33333333 + 4n + n2 = 0

Move the constant term to the right:

Add '27.33333333' to each side of the equation.
-27.33333333 + 4n + 27.33333333 + n2 = 0 + 27.33333333

Reorder the terms:
-27.33333333 + 27.33333333 + 4n + n2 = 0 + 27.33333333

Combine like terms: -27.33333333 + 27.33333333 = 0.00000000
0.00000000 + 4n + n2 = 0 + 27.33333333
4n + n2 = 0 + 27.33333333

Combine like terms: 0 + 27.33333333 = 27.33333333
4n + n2 = 27.33333333

The n term is 4n.  Take half its coefficient (2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
4n + 4 + n2 = 27.33333333 + 4

Reorder the terms:
4 + 4n + n2 = 27.33333333 + 4

Combine like terms: 27.33333333 + 4 = 31.33333333
4 + 4n + n2 = 31.33333333

Factor a perfect square on the left side:
(n + 2)(n + 2) = 31.33333333

Calculate the square root of the right side: 5.597618541

Break this problem into two subproblems by setting 
(n + 2) equal to 5.597618541 and -5.597618541.

Subproblem 1

n + 2 = 5.597618541 Simplifying n + 2 = 5.597618541 Reorder the terms: 2 + n = 5.597618541 Solving 2 + n = 5.597618541 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + n = 5.597618541 + -2 Combine like terms: 2 + -2 = 0 0 + n = 5.597618541 + -2 n = 5.597618541 + -2 Combine like terms: 5.597618541 + -2 = 3.597618541 n = 3.597618541 Simplifying n = 3.597618541

Subproblem 2

n + 2 = -5.597618541 Simplifying n + 2 = -5.597618541 Reorder the terms: 2 + n = -5.597618541 Solving 2 + n = -5.597618541 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + n = -5.597618541 + -2 Combine like terms: 2 + -2 = 0 0 + n = -5.597618541 + -2 n = -5.597618541 + -2 Combine like terms: -5.597618541 + -2 = -7.597618541 n = -7.597618541 Simplifying n = -7.597618541

Solution

The solution to the problem is based on the solutions from the subproblems. n = {3.597618541, -7.597618541}

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