(4n+1)(2n-3)=n

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Solution for (4n+1)(2n-3)=n equation:


Simplifying
(4n + 1)(2n + -3) = n

Reorder the terms:
(1 + 4n)(2n + -3) = n

Reorder the terms:
(1 + 4n)(-3 + 2n) = n

Multiply (1 + 4n) * (-3 + 2n)
(1(-3 + 2n) + 4n * (-3 + 2n)) = n
((-3 * 1 + 2n * 1) + 4n * (-3 + 2n)) = n
((-3 + 2n) + 4n * (-3 + 2n)) = n
(-3 + 2n + (-3 * 4n + 2n * 4n)) = n
(-3 + 2n + (-12n + 8n2)) = n

Combine like terms: 2n + -12n = -10n
(-3 + -10n + 8n2) = n

Solving
-3 + -10n + 8n2 = n

Solving for variable 'n'.

Reorder the terms:
-3 + -10n + -1n + 8n2 = n + -1n

Combine like terms: -10n + -1n = -11n
-3 + -11n + 8n2 = n + -1n

Combine like terms: n + -1n = 0
-3 + -11n + 8n2 = 0

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

Divide each side by '8'.
-0.375 + -1.375n + n2 = 0

Move the constant term to the right:

Add '0.375' to each side of the equation.
-0.375 + -1.375n + 0.375 + n2 = 0 + 0.375

Reorder the terms:
-0.375 + 0.375 + -1.375n + n2 = 0 + 0.375

Combine like terms: -0.375 + 0.375 = 0.000
0.000 + -1.375n + n2 = 0 + 0.375
-1.375n + n2 = 0 + 0.375

Combine like terms: 0 + 0.375 = 0.375
-1.375n + n2 = 0.375

The n term is -1.375n.  Take half its coefficient (-0.6875).
Square it (0.47265625) and add it to both sides.

Add '0.47265625' to each side of the equation.
-1.375n + 0.47265625 + n2 = 0.375 + 0.47265625

Reorder the terms:
0.47265625 + -1.375n + n2 = 0.375 + 0.47265625

Combine like terms: 0.375 + 0.47265625 = 0.84765625
0.47265625 + -1.375n + n2 = 0.84765625

Factor a perfect square on the left side:
(n + -0.6875)(n + -0.6875) = 0.84765625

Calculate the square root of the right side: 0.920682491

Break this problem into two subproblems by setting 
(n + -0.6875) equal to 0.920682491 and -0.920682491.

Subproblem 1

n + -0.6875 = 0.920682491 Simplifying n + -0.6875 = 0.920682491 Reorder the terms: -0.6875 + n = 0.920682491 Solving -0.6875 + n = 0.920682491 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.6875' to each side of the equation. -0.6875 + 0.6875 + n = 0.920682491 + 0.6875 Combine like terms: -0.6875 + 0.6875 = 0.0000 0.0000 + n = 0.920682491 + 0.6875 n = 0.920682491 + 0.6875 Combine like terms: 0.920682491 + 0.6875 = 1.608182491 n = 1.608182491 Simplifying n = 1.608182491

Subproblem 2

n + -0.6875 = -0.920682491 Simplifying n + -0.6875 = -0.920682491 Reorder the terms: -0.6875 + n = -0.920682491 Solving -0.6875 + n = -0.920682491 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.6875' to each side of the equation. -0.6875 + 0.6875 + n = -0.920682491 + 0.6875 Combine like terms: -0.6875 + 0.6875 = 0.0000 0.0000 + n = -0.920682491 + 0.6875 n = -0.920682491 + 0.6875 Combine like terms: -0.920682491 + 0.6875 = -0.233182491 n = -0.233182491 Simplifying n = -0.233182491

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.608182491, -0.233182491}

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