3n^2=-3n+9

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Solution for 3n^2=-3n+9 equation:


Simplifying
3n2 = -3n + 9

Reorder the terms:
3n2 = 9 + -3n

Solving
3n2 = 9 + -3n

Solving for variable 'n'.

Reorder the terms:
-9 + 3n + 3n2 = 9 + -3n + -9 + 3n

Reorder the terms:
-9 + 3n + 3n2 = 9 + -9 + -3n + 3n

Combine like terms: 9 + -9 = 0
-9 + 3n + 3n2 = 0 + -3n + 3n
-9 + 3n + 3n2 = -3n + 3n

Combine like terms: -3n + 3n = 0
-9 + 3n + 3n2 = 0

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

Ignore the factor 3.

Subproblem 1

Set the factor '(-3 + n + n2)' equal to zero and attempt to solve: Simplifying -3 + n + n2 = 0 Solving -3 + n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '3' to each side of the equation. -3 + n + 3 + n2 = 0 + 3 Reorder the terms: -3 + 3 + n + n2 = 0 + 3 Combine like terms: -3 + 3 = 0 0 + n + n2 = 0 + 3 n + n2 = 0 + 3 Combine like terms: 0 + 3 = 3 n + n2 = 3 The n term is n. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. n + 0.25 + n2 = 3 + 0.25 Reorder the terms: 0.25 + n + n2 = 3 + 0.25 Combine like terms: 3 + 0.25 = 3.25 0.25 + n + n2 = 3.25 Factor a perfect square on the left side: (n + 0.5)(n + 0.5) = 3.25 Calculate the square root of the right side: 1.802775638 Break this problem into two subproblems by setting (n + 0.5) equal to 1.802775638 and -1.802775638.

Subproblem 1

n + 0.5 = 1.802775638 Simplifying n + 0.5 = 1.802775638 Reorder the terms: 0.5 + n = 1.802775638 Solving 0.5 + n = 1.802775638 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = 1.802775638 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 1.802775638 + -0.5 n = 1.802775638 + -0.5 Combine like terms: 1.802775638 + -0.5 = 1.302775638 n = 1.302775638 Simplifying n = 1.302775638

Subproblem 2

n + 0.5 = -1.802775638 Simplifying n + 0.5 = -1.802775638 Reorder the terms: 0.5 + n = -1.802775638 Solving 0.5 + n = -1.802775638 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = -1.802775638 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -1.802775638 + -0.5 n = -1.802775638 + -0.5 Combine like terms: -1.802775638 + -0.5 = -2.302775638 n = -2.302775638 Simplifying n = -2.302775638

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.302775638, -2.302775638}

Solution

n = {1.302775638, -2.302775638}

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