n*n+6-3(n)=45

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Solution for n*n+6-3(n)=45 equation:


Simplifying
n * n + 6 + -3(n) = 45

Multiply n * n
n2 + 6 + -3(n) = 45

Reorder the terms:
6 + -3n + n2 = 45

Solving
6 + -3n + n2 = 45

Solving for variable 'n'.

Reorder the terms:
6 + -45 + -3n + n2 = 45 + -45

Combine like terms: 6 + -45 = -39
-39 + -3n + n2 = 45 + -45

Combine like terms: 45 + -45 = 0
-39 + -3n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '39' to each side of the equation.
-39 + -3n + 39 + n2 = 0 + 39

Reorder the terms:
-39 + 39 + -3n + n2 = 0 + 39

Combine like terms: -39 + 39 = 0
0 + -3n + n2 = 0 + 39
-3n + n2 = 0 + 39

Combine like terms: 0 + 39 = 39
-3n + n2 = 39

The n term is -3n.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3n + 2.25 + n2 = 39 + 2.25

Reorder the terms:
2.25 + -3n + n2 = 39 + 2.25

Combine like terms: 39 + 2.25 = 41.25
2.25 + -3n + n2 = 41.25

Factor a perfect square on the left side:
(n + -1.5)(n + -1.5) = 41.25

Calculate the square root of the right side: 6.422616289

Break this problem into two subproblems by setting 
(n + -1.5) equal to 6.422616289 and -6.422616289.

Subproblem 1

n + -1.5 = 6.422616289 Simplifying n + -1.5 = 6.422616289 Reorder the terms: -1.5 + n = 6.422616289 Solving -1.5 + n = 6.422616289 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + n = 6.422616289 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = 6.422616289 + 1.5 n = 6.422616289 + 1.5 Combine like terms: 6.422616289 + 1.5 = 7.922616289 n = 7.922616289 Simplifying n = 7.922616289

Subproblem 2

n + -1.5 = -6.422616289 Simplifying n + -1.5 = -6.422616289 Reorder the terms: -1.5 + n = -6.422616289 Solving -1.5 + n = -6.422616289 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + n = -6.422616289 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = -6.422616289 + 1.5 n = -6.422616289 + 1.5 Combine like terms: -6.422616289 + 1.5 = -4.922616289 n = -4.922616289 Simplifying n = -4.922616289

Solution

The solution to the problem is based on the solutions from the subproblems. n = {7.922616289, -4.922616289}

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