n^2+n=45

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Solution for n^2+n=45 equation:


Simplifying
n2 + n = 45

Reorder the terms:
n + n2 = 45

Solving
n + n2 = 45

Solving for variable 'n'.

Reorder the terms:
-45 + n + n2 = 45 + -45

Combine like terms: 45 + -45 = 0
-45 + n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '45' to each side of the equation.
-45 + n + 45 + n2 = 0 + 45

Reorder the terms:
-45 + 45 + n + n2 = 0 + 45

Combine like terms: -45 + 45 = 0
0 + n + n2 = 0 + 45
n + n2 = 0 + 45

Combine like terms: 0 + 45 = 45
n + n2 = 45

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 = 45 + 0.25

Reorder the terms:
0.25 + n + n2 = 45 + 0.25

Combine like terms: 45 + 0.25 = 45.25
0.25 + n + n2 = 45.25

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 45.25

Calculate the square root of the right side: 6.726812024

Break this problem into two subproblems by setting 
(n + 0.5) equal to 6.726812024 and -6.726812024.

Subproblem 1

n + 0.5 = 6.726812024 Simplifying n + 0.5 = 6.726812024 Reorder the terms: 0.5 + n = 6.726812024 Solving 0.5 + n = 6.726812024 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 = 6.726812024 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 6.726812024 + -0.5 n = 6.726812024 + -0.5 Combine like terms: 6.726812024 + -0.5 = 6.226812024 n = 6.226812024 Simplifying n = 6.226812024

Subproblem 2

n + 0.5 = -6.726812024 Simplifying n + 0.5 = -6.726812024 Reorder the terms: 0.5 + n = -6.726812024 Solving 0.5 + n = -6.726812024 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 = -6.726812024 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -6.726812024 + -0.5 n = -6.726812024 + -0.5 Combine like terms: -6.726812024 + -0.5 = -7.226812024 n = -7.226812024 Simplifying n = -7.226812024

Solution

The solution to the problem is based on the solutions from the subproblems. n = {6.226812024, -7.226812024}

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