5=10n(n-3)-39

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Solution for 5=10n(n-3)-39 equation:


Simplifying
5 = 10n(n + -3) + -39

Reorder the terms:
5 = 10n(-3 + n) + -39
5 = (-3 * 10n + n * 10n) + -39
5 = (-30n + 10n2) + -39

Reorder the terms:
5 = -39 + -30n + 10n2

Solving
5 = -39 + -30n + 10n2

Solving for variable 'n'.

Combine like terms: 5 + 39 = 44
44 + 30n + -10n2 = -39 + -30n + 10n2 + 39 + 30n + -10n2

Reorder the terms:
44 + 30n + -10n2 = -39 + 39 + -30n + 30n + 10n2 + -10n2

Combine like terms: -39 + 39 = 0
44 + 30n + -10n2 = 0 + -30n + 30n + 10n2 + -10n2
44 + 30n + -10n2 = -30n + 30n + 10n2 + -10n2

Combine like terms: -30n + 30n = 0
44 + 30n + -10n2 = 0 + 10n2 + -10n2
44 + 30n + -10n2 = 10n2 + -10n2

Combine like terms: 10n2 + -10n2 = 0
44 + 30n + -10n2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(22 + 15n + -5n2) = 0

Ignore the factor 2.

Subproblem 1

Set the factor '(22 + 15n + -5n2)' equal to zero and attempt to solve: Simplifying 22 + 15n + -5n2 = 0 Solving 22 + 15n + -5n2 = 0 Begin completing the square. Divide all terms by -5 the coefficient of the squared term: Divide each side by '-5'. -4.4 + -3n + n2 = 0 Move the constant term to the right: Add '4.4' to each side of the equation. -4.4 + -3n + 4.4 + n2 = 0 + 4.4 Reorder the terms: -4.4 + 4.4 + -3n + n2 = 0 + 4.4 Combine like terms: -4.4 + 4.4 = 0.0 0.0 + -3n + n2 = 0 + 4.4 -3n + n2 = 0 + 4.4 Combine like terms: 0 + 4.4 = 4.4 -3n + n2 = 4.4 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 = 4.4 + 2.25 Reorder the terms: 2.25 + -3n + n2 = 4.4 + 2.25 Combine like terms: 4.4 + 2.25 = 6.65 2.25 + -3n + n2 = 6.65 Factor a perfect square on the left side: (n + -1.5)(n + -1.5) = 6.65 Calculate the square root of the right side: 2.578759392 Break this problem into two subproblems by setting (n + -1.5) equal to 2.578759392 and -2.578759392.

Subproblem 1

n + -1.5 = 2.578759392 Simplifying n + -1.5 = 2.578759392 Reorder the terms: -1.5 + n = 2.578759392 Solving -1.5 + n = 2.578759392 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 = 2.578759392 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = 2.578759392 + 1.5 n = 2.578759392 + 1.5 Combine like terms: 2.578759392 + 1.5 = 4.078759392 n = 4.078759392 Simplifying n = 4.078759392

Subproblem 2

n + -1.5 = -2.578759392 Simplifying n + -1.5 = -2.578759392 Reorder the terms: -1.5 + n = -2.578759392 Solving -1.5 + n = -2.578759392 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 = -2.578759392 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = -2.578759392 + 1.5 n = -2.578759392 + 1.5 Combine like terms: -2.578759392 + 1.5 = -1.078759392 n = -1.078759392 Simplifying n = -1.078759392

Solution

The solution to the problem is based on the solutions from the subproblems. n = {4.078759392, -1.078759392}

Solution

n = {4.078759392, -1.078759392}

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