1.5n+0.8n(30-n)=54

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Solution for 1.5n+0.8n(30-n)=54 equation:


Simplifying
1.5n + 0.8n(30 + -1n) = 54
1.5n + (30 * 0.8n + -1n * 0.8n) = 54
1.5n + (24n + -0.8n2) = 54

Combine like terms: 1.5n + 24n = 25.5n
25.5n + -0.8n2 = 54

Solving
25.5n + -0.8n2 = 54

Solving for variable 'n'.

Reorder the terms:
-54 + 25.5n + -0.8n2 = 54 + -54

Combine like terms: 54 + -54 = 0
-54 + 25.5n + -0.8n2 = 0

Begin completing the square.  Divide all terms by
-0.8 the coefficient of the squared term: 

Divide each side by '-0.8'.
67.5 + -31.875n + n2 = 0

Move the constant term to the right:

Add '-67.5' to each side of the equation.
67.5 + -31.875n + -67.5 + n2 = 0 + -67.5

Reorder the terms:
67.5 + -67.5 + -31.875n + n2 = 0 + -67.5

Combine like terms: 67.5 + -67.5 = 0.0
0.0 + -31.875n + n2 = 0 + -67.5
-31.875n + n2 = 0 + -67.5

Combine like terms: 0 + -67.5 = -67.5
-31.875n + n2 = -67.5

The n term is -31.875n.  Take half its coefficient (-15.9375).
Square it (254.0039063) and add it to both sides.

Add '254.0039063' to each side of the equation.
-31.875n + 254.0039063 + n2 = -67.5 + 254.0039063

Reorder the terms:
254.0039063 + -31.875n + n2 = -67.5 + 254.0039063

Combine like terms: -67.5 + 254.0039063 = 186.5039063
254.0039063 + -31.875n + n2 = 186.5039063

Factor a perfect square on the left side:
(n + -15.9375)(n + -15.9375) = 186.5039063

Calculate the square root of the right side: 13.656643303

Break this problem into two subproblems by setting 
(n + -15.9375) equal to 13.656643303 and -13.656643303.

Subproblem 1

n + -15.9375 = 13.656643303 Simplifying n + -15.9375 = 13.656643303 Reorder the terms: -15.9375 + n = 13.656643303 Solving -15.9375 + n = 13.656643303 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '15.9375' to each side of the equation. -15.9375 + 15.9375 + n = 13.656643303 + 15.9375 Combine like terms: -15.9375 + 15.9375 = 0.0000 0.0000 + n = 13.656643303 + 15.9375 n = 13.656643303 + 15.9375 Combine like terms: 13.656643303 + 15.9375 = 29.594143303 n = 29.594143303 Simplifying n = 29.594143303

Subproblem 2

n + -15.9375 = -13.656643303 Simplifying n + -15.9375 = -13.656643303 Reorder the terms: -15.9375 + n = -13.656643303 Solving -15.9375 + n = -13.656643303 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '15.9375' to each side of the equation. -15.9375 + 15.9375 + n = -13.656643303 + 15.9375 Combine like terms: -15.9375 + 15.9375 = 0.0000 0.0000 + n = -13.656643303 + 15.9375 n = -13.656643303 + 15.9375 Combine like terms: -13.656643303 + 15.9375 = 2.280856697 n = 2.280856697 Simplifying n = 2.280856697

Solution

The solution to the problem is based on the solutions from the subproblems. n = {29.594143303, 2.280856697}

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