n^2-3n-90=0

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Solution for n^2-3n-90=0 equation:


Simplifying
n2 + -3n + -90 = 0

Reorder the terms:
-90 + -3n + n2 = 0

Solving
-90 + -3n + n2 = 0

Solving for variable 'n'.

Begin completing the square.

Move the constant term to the right:

Add '90' to each side of the equation.
-90 + -3n + 90 + n2 = 0 + 90

Reorder the terms:
-90 + 90 + -3n + n2 = 0 + 90

Combine like terms: -90 + 90 = 0
0 + -3n + n2 = 0 + 90
-3n + n2 = 0 + 90

Combine like terms: 0 + 90 = 90
-3n + n2 = 90

The n term is -3n.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3n + 2.25 + n2 = 90 + 2.25

Reorder the terms:
2.25 + -3n + n2 = 90 + 2.25

Combine like terms: 90 + 2.25 = 92.25
2.25 + -3n + n2 = 92.25

Factor a perfect square on the left side:
(n + -1.5)(n + -1.5) = 92.25

Calculate the square root of the right side: 9.604686356

Break this problem into two subproblems by setting 
(n + -1.5) equal to 9.604686356 and -9.604686356.

Subproblem 1

n + -1.5 = 9.604686356 Simplifying n + -1.5 = 9.604686356 Reorder the terms: -1.5 + n = 9.604686356 Solving -1.5 + n = 9.604686356 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + n = 9.604686356 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = 9.604686356 + 1.5 n = 9.604686356 + 1.5 Combine like terms: 9.604686356 + 1.5 = 11.104686356 n = 11.104686356 Simplifying n = 11.104686356

Subproblem 2

n + -1.5 = -9.604686356 Simplifying n + -1.5 = -9.604686356 Reorder the terms: -1.5 + n = -9.604686356 Solving -1.5 + n = -9.604686356 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + n = -9.604686356 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = -9.604686356 + 1.5 n = -9.604686356 + 1.5 Combine like terms: -9.604686356 + 1.5 = -8.104686356 n = -8.104686356 Simplifying n = -8.104686356

Solution

The solution to the problem is based on the solutions from the subproblems. n = {11.104686356, -8.104686356}

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