3n^2-24n=1

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Solution for 3n^2-24n=1 equation:


Simplifying
3n2 + -24n = 1

Reorder the terms:
-24n + 3n2 = 1

Solving
-24n + 3n2 = 1

Solving for variable 'n'.

Reorder the terms:
-1 + -24n + 3n2 = 1 + -1

Combine like terms: 1 + -1 = 0
-1 + -24n + 3n2 = 0

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

Divide each side by '3'.
-0.3333333333 + -8n + n2 = 0

Move the constant term to the right:

Add '0.3333333333' to each side of the equation.
-0.3333333333 + -8n + 0.3333333333 + n2 = 0 + 0.3333333333

Reorder the terms:
-0.3333333333 + 0.3333333333 + -8n + n2 = 0 + 0.3333333333

Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000
0.0000000000 + -8n + n2 = 0 + 0.3333333333
-8n + n2 = 0 + 0.3333333333

Combine like terms: 0 + 0.3333333333 = 0.3333333333
-8n + n2 = 0.3333333333

The n term is -8n.  Take half its coefficient (-4).
Square it (16) and add it to both sides.

Add '16' to each side of the equation.
-8n + 16 + n2 = 0.3333333333 + 16

Reorder the terms:
16 + -8n + n2 = 0.3333333333 + 16

Combine like terms: 0.3333333333 + 16 = 16.3333333333
16 + -8n + n2 = 16.3333333333

Factor a perfect square on the left side:
(n + -4)(n + -4) = 16.3333333333

Calculate the square root of the right side: 4.041451884

Break this problem into two subproblems by setting 
(n + -4) equal to 4.041451884 and -4.041451884.

Subproblem 1

n + -4 = 4.041451884 Simplifying n + -4 = 4.041451884 Reorder the terms: -4 + n = 4.041451884 Solving -4 + n = 4.041451884 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + n = 4.041451884 + 4 Combine like terms: -4 + 4 = 0 0 + n = 4.041451884 + 4 n = 4.041451884 + 4 Combine like terms: 4.041451884 + 4 = 8.041451884 n = 8.041451884 Simplifying n = 8.041451884

Subproblem 2

n + -4 = -4.041451884 Simplifying n + -4 = -4.041451884 Reorder the terms: -4 + n = -4.041451884 Solving -4 + n = -4.041451884 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + n = -4.041451884 + 4 Combine like terms: -4 + 4 = 0 0 + n = -4.041451884 + 4 n = -4.041451884 + 4 Combine like terms: -4.041451884 + 4 = -0.041451884 n = -0.041451884 Simplifying n = -0.041451884

Solution

The solution to the problem is based on the solutions from the subproblems. n = {8.041451884, -0.041451884}

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