-0.25n^2+6n-27=5

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Solution for -0.25n^2+6n-27=5 equation:


Simplifying
-0.25n2 + 6n + -27 = 5

Reorder the terms:
-27 + 6n + -0.25n2 = 5

Solving
-27 + 6n + -0.25n2 = 5

Solving for variable 'n'.

Reorder the terms:
-27 + -5 + 6n + -0.25n2 = 5 + -5

Combine like terms: -27 + -5 = -32
-32 + 6n + -0.25n2 = 5 + -5

Combine like terms: 5 + -5 = 0
-32 + 6n + -0.25n2 = 0

Begin completing the square.  Divide all terms by
-0.25 the coefficient of the squared term: 

Divide each side by '-0.25'.
128 + -24n + n2 = 0

Move the constant term to the right:

Add '-128' to each side of the equation.
128 + -24n + -128 + n2 = 0 + -128

Reorder the terms:
128 + -128 + -24n + n2 = 0 + -128

Combine like terms: 128 + -128 = 0
0 + -24n + n2 = 0 + -128
-24n + n2 = 0 + -128

Combine like terms: 0 + -128 = -128
-24n + n2 = -128

The n term is -24n.  Take half its coefficient (-12).
Square it (144) and add it to both sides.

Add '144' to each side of the equation.
-24n + 144 + n2 = -128 + 144

Reorder the terms:
144 + -24n + n2 = -128 + 144

Combine like terms: -128 + 144 = 16
144 + -24n + n2 = 16

Factor a perfect square on the left side:
(n + -12)(n + -12) = 16

Calculate the square root of the right side: 4

Break this problem into two subproblems by setting 
(n + -12) equal to 4 and -4.

Subproblem 1

n + -12 = 4 Simplifying n + -12 = 4 Reorder the terms: -12 + n = 4 Solving -12 + n = 4 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '12' to each side of the equation. -12 + 12 + n = 4 + 12 Combine like terms: -12 + 12 = 0 0 + n = 4 + 12 n = 4 + 12 Combine like terms: 4 + 12 = 16 n = 16 Simplifying n = 16

Subproblem 2

n + -12 = -4 Simplifying n + -12 = -4 Reorder the terms: -12 + n = -4 Solving -12 + n = -4 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '12' to each side of the equation. -12 + 12 + n = -4 + 12 Combine like terms: -12 + 12 = 0 0 + n = -4 + 12 n = -4 + 12 Combine like terms: -4 + 12 = 8 n = 8 Simplifying n = 8

Solution

The solution to the problem is based on the solutions from the subproblems. n = {16, 8}

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