27=.5n(n-3)

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Solution for 27=.5n(n-3) equation:


Simplifying
27 = 0.5n(n + -3)

Reorder the terms:
27 = 0.5n(-3 + n)
27 = (-3 * 0.5n + n * 0.5n)
27 = (-1.5n + 0.5n2)

Solving
27 = -1.5n + 0.5n2

Solving for variable 'n'.

Reorder the terms:
27 + 1.5n + -0.5n2 = -1.5n + 1.5n + 0.5n2 + -0.5n2

Combine like terms: -1.5n + 1.5n = 0.0
27 + 1.5n + -0.5n2 = 0.0 + 0.5n2 + -0.5n2
27 + 1.5n + -0.5n2 = 0.5n2 + -0.5n2

Combine like terms: 0.5n2 + -0.5n2 = 0.0
27 + 1.5n + -0.5n2 = 0.0

Begin completing the square.  Divide all terms by
-0.5 the coefficient of the squared term: 

Divide each side by '-0.5'.
-54 + -3n + n2 = 0

Move the constant term to the right:

Add '54' to each side of the equation.
-54 + -3n + 54 + n2 = 0 + 54

Reorder the terms:
-54 + 54 + -3n + n2 = 0 + 54

Combine like terms: -54 + 54 = 0
0 + -3n + n2 = 0 + 54
-3n + n2 = 0 + 54

Combine like terms: 0 + 54 = 54
-3n + n2 = 54

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 = 54 + 2.25

Reorder the terms:
2.25 + -3n + n2 = 54 + 2.25

Combine like terms: 54 + 2.25 = 56.25
2.25 + -3n + n2 = 56.25

Factor a perfect square on the left side:
(n + -1.5)(n + -1.5) = 56.25

Calculate the square root of the right side: 7.5

Break this problem into two subproblems by setting 
(n + -1.5) equal to 7.5 and -7.5.

Subproblem 1

n + -1.5 = 7.5 Simplifying n + -1.5 = 7.5 Reorder the terms: -1.5 + n = 7.5 Solving -1.5 + n = 7.5 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 = 7.5 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = 7.5 + 1.5 n = 7.5 + 1.5 Combine like terms: 7.5 + 1.5 = 9 n = 9 Simplifying n = 9

Subproblem 2

n + -1.5 = -7.5 Simplifying n + -1.5 = -7.5 Reorder the terms: -1.5 + n = -7.5 Solving -1.5 + n = -7.5 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 = -7.5 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = -7.5 + 1.5 n = -7.5 + 1.5 Combine like terms: -7.5 + 1.5 = -6 n = -6 Simplifying n = -6

Solution

The solution to the problem is based on the solutions from the subproblems. n = {9, -6}

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