1.5n+0.8n(38-n)=54

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Solution for 1.5n+0.8n(38-n)=54 equation:


Simplifying
1.5n + 0.8n(38 + -1n) = 54
1.5n + (38 * 0.8n + -1n * 0.8n) = 54
1.5n + (30.4n + -0.8n2) = 54

Combine like terms: 1.5n + 30.4n = 31.9n
31.9n + -0.8n2 = 54

Solving
31.9n + -0.8n2 = 54

Solving for variable 'n'.

Reorder the terms:
-54 + 31.9n + -0.8n2 = 54 + -54

Combine like terms: 54 + -54 = 0
-54 + 31.9n + -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 + -39.875n + n2 = 0

Move the constant term to the right:

Add '-67.5' to each side of the equation.
67.5 + -39.875n + -67.5 + n2 = 0 + -67.5

Reorder the terms:
67.5 + -67.5 + -39.875n + n2 = 0 + -67.5

Combine like terms: 67.5 + -67.5 = 0.0
0.0 + -39.875n + n2 = 0 + -67.5
-39.875n + n2 = 0 + -67.5

Combine like terms: 0 + -67.5 = -67.5
-39.875n + n2 = -67.5

The n term is -39.875n.  Take half its coefficient (-19.9375).
Square it (397.5039063) and add it to both sides.

Add '397.5039063' to each side of the equation.
-39.875n + 397.5039063 + n2 = -67.5 + 397.5039063

Reorder the terms:
397.5039063 + -39.875n + n2 = -67.5 + 397.5039063

Combine like terms: -67.5 + 397.5039063 = 330.0039063
397.5039063 + -39.875n + n2 = 330.0039063

Factor a perfect square on the left side:
(n + -19.9375)(n + -19.9375) = 330.0039063

Calculate the square root of the right side: 18.166009642

Break this problem into two subproblems by setting 
(n + -19.9375) equal to 18.166009642 and -18.166009642.

Subproblem 1

n + -19.9375 = 18.166009642 Simplifying n + -19.9375 = 18.166009642 Reorder the terms: -19.9375 + n = 18.166009642 Solving -19.9375 + n = 18.166009642 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '19.9375' to each side of the equation. -19.9375 + 19.9375 + n = 18.166009642 + 19.9375 Combine like terms: -19.9375 + 19.9375 = 0.0000 0.0000 + n = 18.166009642 + 19.9375 n = 18.166009642 + 19.9375 Combine like terms: 18.166009642 + 19.9375 = 38.103509642 n = 38.103509642 Simplifying n = 38.103509642

Subproblem 2

n + -19.9375 = -18.166009642 Simplifying n + -19.9375 = -18.166009642 Reorder the terms: -19.9375 + n = -18.166009642 Solving -19.9375 + n = -18.166009642 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '19.9375' to each side of the equation. -19.9375 + 19.9375 + n = -18.166009642 + 19.9375 Combine like terms: -19.9375 + 19.9375 = 0.0000 0.0000 + n = -18.166009642 + 19.9375 n = -18.166009642 + 19.9375 Combine like terms: -18.166009642 + 19.9375 = 1.771490358 n = 1.771490358 Simplifying n = 1.771490358

Solution

The solution to the problem is based on the solutions from the subproblems. n = {38.103509642, 1.771490358}

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