10n^2=10n+16

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Solution for 10n^2=10n+16 equation:


Simplifying
10n2 = 10n + 16

Reorder the terms:
10n2 = 16 + 10n

Solving
10n2 = 16 + 10n

Solving for variable 'n'.

Reorder the terms:
-16 + -10n + 10n2 = 16 + 10n + -16 + -10n

Reorder the terms:
-16 + -10n + 10n2 = 16 + -16 + 10n + -10n

Combine like terms: 16 + -16 = 0
-16 + -10n + 10n2 = 0 + 10n + -10n
-16 + -10n + 10n2 = 10n + -10n

Combine like terms: 10n + -10n = 0
-16 + -10n + 10n2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-8 + -5n + 5n2) = 0

Ignore the factor 2.

Subproblem 1

Set the factor '(-8 + -5n + 5n2)' equal to zero and attempt to solve: Simplifying -8 + -5n + 5n2 = 0 Solving -8 + -5n + 5n2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -1.6 + -1n + n2 = 0 Move the constant term to the right: Add '1.6' to each side of the equation. -1.6 + -1n + 1.6 + n2 = 0 + 1.6 Reorder the terms: -1.6 + 1.6 + -1n + n2 = 0 + 1.6 Combine like terms: -1.6 + 1.6 = 0.0 0.0 + -1n + n2 = 0 + 1.6 -1n + n2 = 0 + 1.6 Combine like terms: 0 + 1.6 = 1.6 -1n + n2 = 1.6 The n term is -1n. 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. -1n + 0.25 + n2 = 1.6 + 0.25 Reorder the terms: 0.25 + -1n + n2 = 1.6 + 0.25 Combine like terms: 1.6 + 0.25 = 1.85 0.25 + -1n + n2 = 1.85 Factor a perfect square on the left side: (n + -0.5)(n + -0.5) = 1.85 Calculate the square root of the right side: 1.360147051 Break this problem into two subproblems by setting (n + -0.5) equal to 1.360147051 and -1.360147051.

Subproblem 1

n + -0.5 = 1.360147051 Simplifying n + -0.5 = 1.360147051 Reorder the terms: -0.5 + n = 1.360147051 Solving -0.5 + n = 1.360147051 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 = 1.360147051 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = 1.360147051 + 0.5 n = 1.360147051 + 0.5 Combine like terms: 1.360147051 + 0.5 = 1.860147051 n = 1.860147051 Simplifying n = 1.860147051

Subproblem 2

n + -0.5 = -1.360147051 Simplifying n + -0.5 = -1.360147051 Reorder the terms: -0.5 + n = -1.360147051 Solving -0.5 + n = -1.360147051 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 = -1.360147051 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = -1.360147051 + 0.5 n = -1.360147051 + 0.5 Combine like terms: -1.360147051 + 0.5 = -0.860147051 n = -0.860147051 Simplifying n = -0.860147051

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.860147051, -0.860147051}

Solution

n = {1.860147051, -0.860147051}

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