n^2-3n-18=45

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Solution for n^2-3n-18=45 equation:


Simplifying
n2 + -3n + -18 = 45

Reorder the terms:
-18 + -3n + n2 = 45

Solving
-18 + -3n + n2 = 45

Solving for variable 'n'.

Reorder the terms:
-18 + -45 + -3n + n2 = 45 + -45

Combine like terms: -18 + -45 = -63
-63 + -3n + n2 = 45 + -45

Combine like terms: 45 + -45 = 0
-63 + -3n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '63' to each side of the equation.
-63 + -3n + 63 + n2 = 0 + 63

Reorder the terms:
-63 + 63 + -3n + n2 = 0 + 63

Combine like terms: -63 + 63 = 0
0 + -3n + n2 = 0 + 63
-3n + n2 = 0 + 63

Combine like terms: 0 + 63 = 63
-3n + n2 = 63

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 = 63 + 2.25

Reorder the terms:
2.25 + -3n + n2 = 63 + 2.25

Combine like terms: 63 + 2.25 = 65.25
2.25 + -3n + n2 = 65.25

Factor a perfect square on the left side:
(n + -1.5)(n + -1.5) = 65.25

Calculate the square root of the right side: 8.077747211

Break this problem into two subproblems by setting 
(n + -1.5) equal to 8.077747211 and -8.077747211.

Subproblem 1

n + -1.5 = 8.077747211 Simplifying n + -1.5 = 8.077747211 Reorder the terms: -1.5 + n = 8.077747211 Solving -1.5 + n = 8.077747211 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 = 8.077747211 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = 8.077747211 + 1.5 n = 8.077747211 + 1.5 Combine like terms: 8.077747211 + 1.5 = 9.577747211 n = 9.577747211 Simplifying n = 9.577747211

Subproblem 2

n + -1.5 = -8.077747211 Simplifying n + -1.5 = -8.077747211 Reorder the terms: -1.5 + n = -8.077747211 Solving -1.5 + n = -8.077747211 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 = -8.077747211 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = -8.077747211 + 1.5 n = -8.077747211 + 1.5 Combine like terms: -8.077747211 + 1.5 = -6.577747211 n = -6.577747211 Simplifying n = -6.577747211

Solution

The solution to the problem is based on the solutions from the subproblems. n = {9.577747211, -6.577747211}

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