n^2+23n-100=0

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Solution for n^2+23n-100=0 equation:


Simplifying
n2 + 23n + -100 = 0

Reorder the terms:
-100 + 23n + n2 = 0

Solving
-100 + 23n + n2 = 0

Solving for variable 'n'.

Begin completing the square.

Move the constant term to the right:

Add '100' to each side of the equation.
-100 + 23n + 100 + n2 = 0 + 100

Reorder the terms:
-100 + 100 + 23n + n2 = 0 + 100

Combine like terms: -100 + 100 = 0
0 + 23n + n2 = 0 + 100
23n + n2 = 0 + 100

Combine like terms: 0 + 100 = 100
23n + n2 = 100

The n term is 23n.  Take half its coefficient (11.5).
Square it (132.25) and add it to both sides.

Add '132.25' to each side of the equation.
23n + 132.25 + n2 = 100 + 132.25

Reorder the terms:
132.25 + 23n + n2 = 100 + 132.25

Combine like terms: 100 + 132.25 = 232.25
132.25 + 23n + n2 = 232.25

Factor a perfect square on the left side:
(n + 11.5)(n + 11.5) = 232.25

Calculate the square root of the right side: 15.239750654

Break this problem into two subproblems by setting 
(n + 11.5) equal to 15.239750654 and -15.239750654.

Subproblem 1

n + 11.5 = 15.239750654 Simplifying n + 11.5 = 15.239750654 Reorder the terms: 11.5 + n = 15.239750654 Solving 11.5 + n = 15.239750654 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-11.5' to each side of the equation. 11.5 + -11.5 + n = 15.239750654 + -11.5 Combine like terms: 11.5 + -11.5 = 0.0 0.0 + n = 15.239750654 + -11.5 n = 15.239750654 + -11.5 Combine like terms: 15.239750654 + -11.5 = 3.739750654 n = 3.739750654 Simplifying n = 3.739750654

Subproblem 2

n + 11.5 = -15.239750654 Simplifying n + 11.5 = -15.239750654 Reorder the terms: 11.5 + n = -15.239750654 Solving 11.5 + n = -15.239750654 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-11.5' to each side of the equation. 11.5 + -11.5 + n = -15.239750654 + -11.5 Combine like terms: 11.5 + -11.5 = 0.0 0.0 + n = -15.239750654 + -11.5 n = -15.239750654 + -11.5 Combine like terms: -15.239750654 + -11.5 = -26.739750654 n = -26.739750654 Simplifying n = -26.739750654

Solution

The solution to the problem is based on the solutions from the subproblems. n = {3.739750654, -26.739750654}

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