10n^2=20-8n

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Solution for 10n^2=20-8n equation:


Simplifying
10n2 = 20 + -8n

Solving
10n2 = 20 + -8n

Solving for variable 'n'.

Reorder the terms:
-20 + 8n + 10n2 = 20 + -8n + -20 + 8n

Reorder the terms:
-20 + 8n + 10n2 = 20 + -20 + -8n + 8n

Combine like terms: 20 + -20 = 0
-20 + 8n + 10n2 = 0 + -8n + 8n
-20 + 8n + 10n2 = -8n + 8n

Combine like terms: -8n + 8n = 0
-20 + 8n + 10n2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-10 + 4n + 5n2) = 0

Ignore the factor 2.

Subproblem 1

Set the factor '(-10 + 4n + 5n2)' equal to zero and attempt to solve: Simplifying -10 + 4n + 5n2 = 0 Solving -10 + 4n + 5n2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -2 + 0.8n + n2 = 0 Move the constant term to the right: Add '2' to each side of the equation. -2 + 0.8n + 2 + n2 = 0 + 2 Reorder the terms: -2 + 2 + 0.8n + n2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + 0.8n + n2 = 0 + 2 0.8n + n2 = 0 + 2 Combine like terms: 0 + 2 = 2 0.8n + n2 = 2 The n term is 0.8n. Take half its coefficient (0.4). Square it (0.16) and add it to both sides. Add '0.16' to each side of the equation. 0.8n + 0.16 + n2 = 2 + 0.16 Reorder the terms: 0.16 + 0.8n + n2 = 2 + 0.16 Combine like terms: 2 + 0.16 = 2.16 0.16 + 0.8n + n2 = 2.16 Factor a perfect square on the left side: (n + 0.4)(n + 0.4) = 2.16 Calculate the square root of the right side: 1.469693846 Break this problem into two subproblems by setting (n + 0.4) equal to 1.469693846 and -1.469693846.

Subproblem 1

n + 0.4 = 1.469693846 Simplifying n + 0.4 = 1.469693846 Reorder the terms: 0.4 + n = 1.469693846 Solving 0.4 + n = 1.469693846 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.4' to each side of the equation. 0.4 + -0.4 + n = 1.469693846 + -0.4 Combine like terms: 0.4 + -0.4 = 0.0 0.0 + n = 1.469693846 + -0.4 n = 1.469693846 + -0.4 Combine like terms: 1.469693846 + -0.4 = 1.069693846 n = 1.069693846 Simplifying n = 1.069693846

Subproblem 2

n + 0.4 = -1.469693846 Simplifying n + 0.4 = -1.469693846 Reorder the terms: 0.4 + n = -1.469693846 Solving 0.4 + n = -1.469693846 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.4' to each side of the equation. 0.4 + -0.4 + n = -1.469693846 + -0.4 Combine like terms: 0.4 + -0.4 = 0.0 0.0 + n = -1.469693846 + -0.4 n = -1.469693846 + -0.4 Combine like terms: -1.469693846 + -0.4 = -1.869693846 n = -1.869693846 Simplifying n = -1.869693846

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.069693846, -1.869693846}

Solution

n = {1.069693846, -1.869693846}

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