0.5n^2+1.5n=9

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Solution for 0.5n^2+1.5n=9 equation:


Simplifying
0.5n2 + 1.5n = 9

Reorder the terms:
1.5n + 0.5n2 = 9

Solving
1.5n + 0.5n2 = 9

Solving for variable 'n'.

Reorder the terms:
-9 + 1.5n + 0.5n2 = 9 + -9

Combine like terms: 9 + -9 = 0
-9 + 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'.
-18 + 3n + n2 = 0

Move the constant term to the right:

Add '18' to each side of the equation.
-18 + 3n + 18 + n2 = 0 + 18

Reorder the terms:
-18 + 18 + 3n + n2 = 0 + 18

Combine like terms: -18 + 18 = 0
0 + 3n + n2 = 0 + 18
3n + n2 = 0 + 18

Combine like terms: 0 + 18 = 18
3n + n2 = 18

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 = 18 + 2.25

Reorder the terms:
2.25 + 3n + n2 = 18 + 2.25

Combine like terms: 18 + 2.25 = 20.25
2.25 + 3n + n2 = 20.25

Factor a perfect square on the left side:
(n + 1.5)(n + 1.5) = 20.25

Calculate the square root of the right side: 4.5

Break this problem into two subproblems by setting 
(n + 1.5) equal to 4.5 and -4.5.

Subproblem 1

n + 1.5 = 4.5 Simplifying n + 1.5 = 4.5 Reorder the terms: 1.5 + n = 4.5 Solving 1.5 + n = 4.5 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 = 4.5 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = 4.5 + -1.5 n = 4.5 + -1.5 Combine like terms: 4.5 + -1.5 = 3 n = 3 Simplifying n = 3

Subproblem 2

n + 1.5 = -4.5 Simplifying n + 1.5 = -4.5 Reorder the terms: 1.5 + n = -4.5 Solving 1.5 + n = -4.5 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 = -4.5 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = -4.5 + -1.5 n = -4.5 + -1.5 Combine like terms: -4.5 + -1.5 = -6 n = -6 Simplifying n = -6

Solution

The solution to the problem is based on the solutions from the subproblems. n = {3, -6}

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