3n^2+24n-84=0

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Solution for 3n^2+24n-84=0 equation:


Simplifying
3n2 + 24n + -84 = 0

Reorder the terms:
-84 + 24n + 3n2 = 0

Solving
-84 + 24n + 3n2 = 0

Solving for variable 'n'.

Factor out the Greatest Common Factor (GCF), '3'.
3(-28 + 8n + n2) = 0

Ignore the factor 3.

Subproblem 1

Set the factor '(-28 + 8n + n2)' equal to zero and attempt to solve: Simplifying -28 + 8n + n2 = 0 Solving -28 + 8n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '28' to each side of the equation. -28 + 8n + 28 + n2 = 0 + 28 Reorder the terms: -28 + 28 + 8n + n2 = 0 + 28 Combine like terms: -28 + 28 = 0 0 + 8n + n2 = 0 + 28 8n + n2 = 0 + 28 Combine like terms: 0 + 28 = 28 8n + n2 = 28 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 = 28 + 16 Reorder the terms: 16 + 8n + n2 = 28 + 16 Combine like terms: 28 + 16 = 44 16 + 8n + n2 = 44 Factor a perfect square on the left side: (n + 4)(n + 4) = 44 Calculate the square root of the right side: 6.633249581 Break this problem into two subproblems by setting (n + 4) equal to 6.633249581 and -6.633249581.

Subproblem 1

n + 4 = 6.633249581 Simplifying n + 4 = 6.633249581 Reorder the terms: 4 + n = 6.633249581 Solving 4 + n = 6.633249581 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 = 6.633249581 + -4 Combine like terms: 4 + -4 = 0 0 + n = 6.633249581 + -4 n = 6.633249581 + -4 Combine like terms: 6.633249581 + -4 = 2.633249581 n = 2.633249581 Simplifying n = 2.633249581

Subproblem 2

n + 4 = -6.633249581 Simplifying n + 4 = -6.633249581 Reorder the terms: 4 + n = -6.633249581 Solving 4 + n = -6.633249581 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 = -6.633249581 + -4 Combine like terms: 4 + -4 = 0 0 + n = -6.633249581 + -4 n = -6.633249581 + -4 Combine like terms: -6.633249581 + -4 = -10.633249581 n = -10.633249581 Simplifying n = -10.633249581

Solution

The solution to the problem is based on the solutions from the subproblems. n = {2.633249581, -10.633249581}

Solution

n = {2.633249581, -10.633249581}

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