6n^2=18n+36

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Solution for 6n^2=18n+36 equation:


Simplifying
6n2 = 18n + 36

Reorder the terms:
6n2 = 36 + 18n

Solving
6n2 = 36 + 18n

Solving for variable 'n'.

Reorder the terms:
-36 + -18n + 6n2 = 36 + 18n + -36 + -18n

Reorder the terms:
-36 + -18n + 6n2 = 36 + -36 + 18n + -18n

Combine like terms: 36 + -36 = 0
-36 + -18n + 6n2 = 0 + 18n + -18n
-36 + -18n + 6n2 = 18n + -18n

Combine like terms: 18n + -18n = 0
-36 + -18n + 6n2 = 0

Factor out the Greatest Common Factor (GCF), '6'.
6(-6 + -3n + n2) = 0

Ignore the factor 6.

Subproblem 1

Set the factor '(-6 + -3n + n2)' equal to zero and attempt to solve: Simplifying -6 + -3n + n2 = 0 Solving -6 + -3n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '6' to each side of the equation. -6 + -3n + 6 + n2 = 0 + 6 Reorder the terms: -6 + 6 + -3n + n2 = 0 + 6 Combine like terms: -6 + 6 = 0 0 + -3n + n2 = 0 + 6 -3n + n2 = 0 + 6 Combine like terms: 0 + 6 = 6 -3n + n2 = 6 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 = 6 + 2.25 Reorder the terms: 2.25 + -3n + n2 = 6 + 2.25 Combine like terms: 6 + 2.25 = 8.25 2.25 + -3n + n2 = 8.25 Factor a perfect square on the left side: (n + -1.5)(n + -1.5) = 8.25 Calculate the square root of the right side: 2.872281323 Break this problem into two subproblems by setting (n + -1.5) equal to 2.872281323 and -2.872281323.

Subproblem 1

n + -1.5 = 2.872281323 Simplifying n + -1.5 = 2.872281323 Reorder the terms: -1.5 + n = 2.872281323 Solving -1.5 + n = 2.872281323 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 = 2.872281323 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = 2.872281323 + 1.5 n = 2.872281323 + 1.5 Combine like terms: 2.872281323 + 1.5 = 4.372281323 n = 4.372281323 Simplifying n = 4.372281323

Subproblem 2

n + -1.5 = -2.872281323 Simplifying n + -1.5 = -2.872281323 Reorder the terms: -1.5 + n = -2.872281323 Solving -1.5 + n = -2.872281323 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 = -2.872281323 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = -2.872281323 + 1.5 n = -2.872281323 + 1.5 Combine like terms: -2.872281323 + 1.5 = -1.372281323 n = -1.372281323 Simplifying n = -1.372281323

Solution

The solution to the problem is based on the solutions from the subproblems. n = {4.372281323, -1.372281323}

Solution

n = {4.372281323, -1.372281323}

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