n^2+n=40

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Solution for n^2+n=40 equation:


Simplifying
n2 + n = 40

Reorder the terms:
n + n2 = 40

Solving
n + n2 = 40

Solving for variable 'n'.

Reorder the terms:
-40 + n + n2 = 40 + -40

Combine like terms: 40 + -40 = 0
-40 + n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '40' to each side of the equation.
-40 + n + 40 + n2 = 0 + 40

Reorder the terms:
-40 + 40 + n + n2 = 0 + 40

Combine like terms: -40 + 40 = 0
0 + n + n2 = 0 + 40
n + n2 = 0 + 40

Combine like terms: 0 + 40 = 40
n + n2 = 40

The n term is n.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
n + 0.25 + n2 = 40 + 0.25

Reorder the terms:
0.25 + n + n2 = 40 + 0.25

Combine like terms: 40 + 0.25 = 40.25
0.25 + n + n2 = 40.25

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 40.25

Calculate the square root of the right side: 6.34428877

Break this problem into two subproblems by setting 
(n + 0.5) equal to 6.34428877 and -6.34428877.

Subproblem 1

n + 0.5 = 6.34428877 Simplifying n + 0.5 = 6.34428877 Reorder the terms: 0.5 + n = 6.34428877 Solving 0.5 + n = 6.34428877 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = 6.34428877 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 6.34428877 + -0.5 n = 6.34428877 + -0.5 Combine like terms: 6.34428877 + -0.5 = 5.84428877 n = 5.84428877 Simplifying n = 5.84428877

Subproblem 2

n + 0.5 = -6.34428877 Simplifying n + 0.5 = -6.34428877 Reorder the terms: 0.5 + n = -6.34428877 Solving 0.5 + n = -6.34428877 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = -6.34428877 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -6.34428877 + -0.5 n = -6.34428877 + -0.5 Combine like terms: -6.34428877 + -0.5 = -6.84428877 n = -6.84428877 Simplifying n = -6.84428877

Solution

The solution to the problem is based on the solutions from the subproblems. n = {5.84428877, -6.84428877}

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