n^2-n-20=45

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Solution for n^2-n-20=45 equation:


Simplifying
n2 + -1n + -20 = 45

Reorder the terms:
-20 + -1n + n2 = 45

Solving
-20 + -1n + n2 = 45

Solving for variable 'n'.

Reorder the terms:
-20 + -45 + -1n + n2 = 45 + -45

Combine like terms: -20 + -45 = -65
-65 + -1n + n2 = 45 + -45

Combine like terms: 45 + -45 = 0
-65 + -1n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '65' to each side of the equation.
-65 + -1n + 65 + n2 = 0 + 65

Reorder the terms:
-65 + 65 + -1n + n2 = 0 + 65

Combine like terms: -65 + 65 = 0
0 + -1n + n2 = 0 + 65
-1n + n2 = 0 + 65

Combine like terms: 0 + 65 = 65
-1n + n2 = 65

The n term is -1n.  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.
-1n + 0.25 + n2 = 65 + 0.25

Reorder the terms:
0.25 + -1n + n2 = 65 + 0.25

Combine like terms: 65 + 0.25 = 65.25
0.25 + -1n + n2 = 65.25

Factor a perfect square on the left side:
(n + -0.5)(n + -0.5) = 65.25

Calculate the square root of the right side: 8.077747211

Break this problem into two subproblems by setting 
(n + -0.5) equal to 8.077747211 and -8.077747211.

Subproblem 1

n + -0.5 = 8.077747211 Simplifying n + -0.5 = 8.077747211 Reorder the terms: -0.5 + n = 8.077747211 Solving -0.5 + n = 8.077747211 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 = 8.077747211 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = 8.077747211 + 0.5 n = 8.077747211 + 0.5 Combine like terms: 8.077747211 + 0.5 = 8.577747211 n = 8.577747211 Simplifying n = 8.577747211

Subproblem 2

n + -0.5 = -8.077747211 Simplifying n + -0.5 = -8.077747211 Reorder the terms: -0.5 + n = -8.077747211 Solving -0.5 + n = -8.077747211 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 = -8.077747211 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = -8.077747211 + 0.5 n = -8.077747211 + 0.5 Combine like terms: -8.077747211 + 0.5 = -7.577747211 n = -7.577747211 Simplifying n = -7.577747211

Solution

The solution to the problem is based on the solutions from the subproblems. n = {8.577747211, -7.577747211}

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