n^2-3n-135=0

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Solution for n^2-3n-135=0 equation:


Simplifying
n2 + -3n + -135 = 0

Reorder the terms:
-135 + -3n + n2 = 0

Solving
-135 + -3n + n2 = 0

Solving for variable 'n'.

Begin completing the square.

Move the constant term to the right:

Add '135' to each side of the equation.
-135 + -3n + 135 + n2 = 0 + 135

Reorder the terms:
-135 + 135 + -3n + n2 = 0 + 135

Combine like terms: -135 + 135 = 0
0 + -3n + n2 = 0 + 135
-3n + n2 = 0 + 135

Combine like terms: 0 + 135 = 135
-3n + n2 = 135

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 = 135 + 2.25

Reorder the terms:
2.25 + -3n + n2 = 135 + 2.25

Combine like terms: 135 + 2.25 = 137.25
2.25 + -3n + n2 = 137.25

Factor a perfect square on the left side:
(n + -1.5)(n + -1.5) = 137.25

Calculate the square root of the right side: 11.715374514

Break this problem into two subproblems by setting 
(n + -1.5) equal to 11.715374514 and -11.715374514.

Subproblem 1

n + -1.5 = 11.715374514 Simplifying n + -1.5 = 11.715374514 Reorder the terms: -1.5 + n = 11.715374514 Solving -1.5 + n = 11.715374514 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 = 11.715374514 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = 11.715374514 + 1.5 n = 11.715374514 + 1.5 Combine like terms: 11.715374514 + 1.5 = 13.215374514 n = 13.215374514 Simplifying n = 13.215374514

Subproblem 2

n + -1.5 = -11.715374514 Simplifying n + -1.5 = -11.715374514 Reorder the terms: -1.5 + n = -11.715374514 Solving -1.5 + n = -11.715374514 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 = -11.715374514 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = -11.715374514 + 1.5 n = -11.715374514 + 1.5 Combine like terms: -11.715374514 + 1.5 = -10.215374514 n = -10.215374514 Simplifying n = -10.215374514

Solution

The solution to the problem is based on the solutions from the subproblems. n = {13.215374514, -10.215374514}

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