n^2-3n-238=1

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Solution for n^2-3n-238=1 equation:


Simplifying
n2 + -3n + -238 = 1

Reorder the terms:
-238 + -3n + n2 = 1

Solving
-238 + -3n + n2 = 1

Solving for variable 'n'.

Reorder the terms:
-238 + -1 + -3n + n2 = 1 + -1

Combine like terms: -238 + -1 = -239
-239 + -3n + n2 = 1 + -1

Combine like terms: 1 + -1 = 0
-239 + -3n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '239' to each side of the equation.
-239 + -3n + 239 + n2 = 0 + 239

Reorder the terms:
-239 + 239 + -3n + n2 = 0 + 239

Combine like terms: -239 + 239 = 0
0 + -3n + n2 = 0 + 239
-3n + n2 = 0 + 239

Combine like terms: 0 + 239 = 239
-3n + n2 = 239

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 = 239 + 2.25

Reorder the terms:
2.25 + -3n + n2 = 239 + 2.25

Combine like terms: 239 + 2.25 = 241.25
2.25 + -3n + n2 = 241.25

Factor a perfect square on the left side:
(n + -1.5)(n + -1.5) = 241.25

Calculate the square root of the right side: 15.532224567

Break this problem into two subproblems by setting 
(n + -1.5) equal to 15.532224567 and -15.532224567.

Subproblem 1

n + -1.5 = 15.532224567 Simplifying n + -1.5 = 15.532224567 Reorder the terms: -1.5 + n = 15.532224567 Solving -1.5 + n = 15.532224567 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 = 15.532224567 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = 15.532224567 + 1.5 n = 15.532224567 + 1.5 Combine like terms: 15.532224567 + 1.5 = 17.032224567 n = 17.032224567 Simplifying n = 17.032224567

Subproblem 2

n + -1.5 = -15.532224567 Simplifying n + -1.5 = -15.532224567 Reorder the terms: -1.5 + n = -15.532224567 Solving -1.5 + n = -15.532224567 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 = -15.532224567 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = -15.532224567 + 1.5 n = -15.532224567 + 1.5 Combine like terms: -15.532224567 + 1.5 = -14.032224567 n = -14.032224567 Simplifying n = -14.032224567

Solution

The solution to the problem is based on the solutions from the subproblems. n = {17.032224567, -14.032224567}

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