3n^2-24n+6=0

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Solution for 3n^2-24n+6=0 equation:


Simplifying
3n2 + -24n + 6 = 0

Reorder the terms:
6 + -24n + 3n2 = 0

Solving
6 + -24n + 3n2 = 0

Solving for variable 'n'.

Factor out the Greatest Common Factor (GCF), '3'.
3(2 + -8n + n2) = 0

Ignore the factor 3.

Subproblem 1

Set the factor '(2 + -8n + n2)' equal to zero and attempt to solve: Simplifying 2 + -8n + n2 = 0 Solving 2 + -8n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '-2' to each side of the equation. 2 + -8n + -2 + n2 = 0 + -2 Reorder the terms: 2 + -2 + -8n + n2 = 0 + -2 Combine like terms: 2 + -2 = 0 0 + -8n + n2 = 0 + -2 -8n + n2 = 0 + -2 Combine like terms: 0 + -2 = -2 -8n + n2 = -2 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 = -2 + 16 Reorder the terms: 16 + -8n + n2 = -2 + 16 Combine like terms: -2 + 16 = 14 16 + -8n + n2 = 14 Factor a perfect square on the left side: (n + -4)(n + -4) = 14 Calculate the square root of the right side: 3.741657387 Break this problem into two subproblems by setting (n + -4) equal to 3.741657387 and -3.741657387.

Subproblem 1

n + -4 = 3.741657387 Simplifying n + -4 = 3.741657387 Reorder the terms: -4 + n = 3.741657387 Solving -4 + n = 3.741657387 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 = 3.741657387 + 4 Combine like terms: -4 + 4 = 0 0 + n = 3.741657387 + 4 n = 3.741657387 + 4 Combine like terms: 3.741657387 + 4 = 7.741657387 n = 7.741657387 Simplifying n = 7.741657387

Subproblem 2

n + -4 = -3.741657387 Simplifying n + -4 = -3.741657387 Reorder the terms: -4 + n = -3.741657387 Solving -4 + n = -3.741657387 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 = -3.741657387 + 4 Combine like terms: -4 + 4 = 0 0 + n = -3.741657387 + 4 n = -3.741657387 + 4 Combine like terms: -3.741657387 + 4 = 0.258342613 n = 0.258342613 Simplifying n = 0.258342613

Solution

The solution to the problem is based on the solutions from the subproblems. n = {7.741657387, 0.258342613}

Solution

n = {7.741657387, 0.258342613}

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