0.5*(n+2)*(n-3)=12

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Solution for 0.5*(n+2)*(n-3)=12 equation:


Simplifying
0.5(n + 2)(n + -3) = 12

Reorder the terms:
0.5(2 + n)(n + -3) = 12

Reorder the terms:
0.5(2 + n)(-3 + n) = 12

Multiply (2 + n) * (-3 + n)
0.5(2(-3 + n) + n(-3 + n)) = 12
0.5((-3 * 2 + n * 2) + n(-3 + n)) = 12
0.5((-6 + 2n) + n(-3 + n)) = 12
0.5(-6 + 2n + (-3 * n + n * n)) = 12
0.5(-6 + 2n + (-3n + n2)) = 12

Combine like terms: 2n + -3n = -1n
0.5(-6 + -1n + n2) = 12
(-6 * 0.5 + -1n * 0.5 + n2 * 0.5) = 12
(-3 + -0.5n + 0.5n2) = 12

Solving
-3 + -0.5n + 0.5n2 = 12

Solving for variable 'n'.

Reorder the terms:
-3 + -12 + -0.5n + 0.5n2 = 12 + -12

Combine like terms: -3 + -12 = -15
-15 + -0.5n + 0.5n2 = 12 + -12

Combine like terms: 12 + -12 = 0
-15 + -0.5n + 0.5n2 = 0

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

Divide each side by '0.5'.
-30 + -1n + n2 = 0

Move the constant term to the right:

Add '30' to each side of the equation.
-30 + -1n + 30 + n2 = 0 + 30

Reorder the terms:
-30 + 30 + -1n + n2 = 0 + 30

Combine like terms: -30 + 30 = 0
0 + -1n + n2 = 0 + 30
-1n + n2 = 0 + 30

Combine like terms: 0 + 30 = 30
-1n + n2 = 30

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 = 30 + 0.25

Reorder the terms:
0.25 + -1n + n2 = 30 + 0.25

Combine like terms: 30 + 0.25 = 30.25
0.25 + -1n + n2 = 30.25

Factor a perfect square on the left side:
(n + -0.5)(n + -0.5) = 30.25

Calculate the square root of the right side: 5.5

Break this problem into two subproblems by setting 
(n + -0.5) equal to 5.5 and -5.5.

Subproblem 1

n + -0.5 = 5.5 Simplifying n + -0.5 = 5.5 Reorder the terms: -0.5 + n = 5.5 Solving -0.5 + n = 5.5 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 = 5.5 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = 5.5 + 0.5 n = 5.5 + 0.5 Combine like terms: 5.5 + 0.5 = 6 n = 6 Simplifying n = 6

Subproblem 2

n + -0.5 = -5.5 Simplifying n + -0.5 = -5.5 Reorder the terms: -0.5 + n = -5.5 Solving -0.5 + n = -5.5 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 = -5.5 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = -5.5 + 0.5 n = -5.5 + 0.5 Combine like terms: -5.5 + 0.5 = -5 n = -5 Simplifying n = -5

Solution

The solution to the problem is based on the solutions from the subproblems. n = {6, -5}

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