0.5n(n-3)=20

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Solution for 0.5n(n-3)=20 equation:


Simplifying
0.5n(n + -3) = 20

Reorder the terms:
0.5n(-3 + n) = 20
(-3 * 0.5n + n * 0.5n) = 20
(-1.5n + 0.5n2) = 20

Solving
-1.5n + 0.5n2 = 20

Solving for variable 'n'.

Reorder the terms:
-20 + -1.5n + 0.5n2 = 20 + -20

Combine like terms: 20 + -20 = 0
-20 + -1.5n + 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'.
-40 + -3n + n2 = 0

Move the constant term to the right:

Add '40' to each side of the equation.
-40 + -3n + 40 + n2 = 0 + 40

Reorder the terms:
-40 + 40 + -3n + n2 = 0 + 40

Combine like terms: -40 + 40 = 0
0 + -3n + n2 = 0 + 40
-3n + n2 = 0 + 40

Combine like terms: 0 + 40 = 40
-3n + n2 = 40

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 = 40 + 2.25

Reorder the terms:
2.25 + -3n + n2 = 40 + 2.25

Combine like terms: 40 + 2.25 = 42.25
2.25 + -3n + n2 = 42.25

Factor a perfect square on the left side:
(n + -1.5)(n + -1.5) = 42.25

Calculate the square root of the right side: 6.5

Break this problem into two subproblems by setting 
(n + -1.5) equal to 6.5 and -6.5.

Subproblem 1

n + -1.5 = 6.5 Simplifying n + -1.5 = 6.5 Reorder the terms: -1.5 + n = 6.5 Solving -1.5 + n = 6.5 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 = 6.5 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = 6.5 + 1.5 n = 6.5 + 1.5 Combine like terms: 6.5 + 1.5 = 8 n = 8 Simplifying n = 8

Subproblem 2

n + -1.5 = -6.5 Simplifying n + -1.5 = -6.5 Reorder the terms: -1.5 + n = -6.5 Solving -1.5 + n = -6.5 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 = -6.5 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = -6.5 + 1.5 n = -6.5 + 1.5 Combine like terms: -6.5 + 1.5 = -5 n = -5 Simplifying n = -5

Solution

The solution to the problem is based on the solutions from the subproblems. n = {8, -5}

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