n^2+9n-1000=0

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Solution for n^2+9n-1000=0 equation:


Simplifying
n2 + 9n + -1000 = 0

Reorder the terms:
-1000 + 9n + n2 = 0

Solving
-1000 + 9n + n2 = 0

Solving for variable 'n'.

Begin completing the square.

Move the constant term to the right:

Add '1000' to each side of the equation.
-1000 + 9n + 1000 + n2 = 0 + 1000

Reorder the terms:
-1000 + 1000 + 9n + n2 = 0 + 1000

Combine like terms: -1000 + 1000 = 0
0 + 9n + n2 = 0 + 1000
9n + n2 = 0 + 1000

Combine like terms: 0 + 1000 = 1000
9n + n2 = 1000

The n term is 9n.  Take half its coefficient (4.5).
Square it (20.25) and add it to both sides.

Add '20.25' to each side of the equation.
9n + 20.25 + n2 = 1000 + 20.25

Reorder the terms:
20.25 + 9n + n2 = 1000 + 20.25

Combine like terms: 1000 + 20.25 = 1020.25
20.25 + 9n + n2 = 1020.25

Factor a perfect square on the left side:
(n + 4.5)(n + 4.5) = 1020.25

Calculate the square root of the right side: 31.941352507

Break this problem into two subproblems by setting 
(n + 4.5) equal to 31.941352507 and -31.941352507.

Subproblem 1

n + 4.5 = 31.941352507 Simplifying n + 4.5 = 31.941352507 Reorder the terms: 4.5 + n = 31.941352507 Solving 4.5 + n = 31.941352507 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-4.5' to each side of the equation. 4.5 + -4.5 + n = 31.941352507 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + n = 31.941352507 + -4.5 n = 31.941352507 + -4.5 Combine like terms: 31.941352507 + -4.5 = 27.441352507 n = 27.441352507 Simplifying n = 27.441352507

Subproblem 2

n + 4.5 = -31.941352507 Simplifying n + 4.5 = -31.941352507 Reorder the terms: 4.5 + n = -31.941352507 Solving 4.5 + n = -31.941352507 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-4.5' to each side of the equation. 4.5 + -4.5 + n = -31.941352507 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + n = -31.941352507 + -4.5 n = -31.941352507 + -4.5 Combine like terms: -31.941352507 + -4.5 = -36.441352507 n = -36.441352507 Simplifying n = -36.441352507

Solution

The solution to the problem is based on the solutions from the subproblems. n = {27.441352507, -36.441352507}

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