5n^2+8n=10

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Solution for 5n^2+8n=10 equation:


Simplifying
5n2 + 8n = 10

Reorder the terms:
8n + 5n2 = 10

Solving
8n + 5n2 = 10

Solving for variable 'n'.

Reorder the terms:
-10 + 8n + 5n2 = 10 + -10

Combine like terms: 10 + -10 = 0
-10 + 8n + 5n2 = 0

Begin completing the square.  Divide all terms by
5 the coefficient of the squared term: 

Divide each side by '5'.
-2 + 1.6n + n2 = 0

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + 1.6n + 2 + n2 = 0 + 2

Reorder the terms:
-2 + 2 + 1.6n + n2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + 1.6n + n2 = 0 + 2
1.6n + n2 = 0 + 2

Combine like terms: 0 + 2 = 2
1.6n + n2 = 2

The n term is 1.6n.  Take half its coefficient (0.8).
Square it (0.64) and add it to both sides.

Add '0.64' to each side of the equation.
1.6n + 0.64 + n2 = 2 + 0.64

Reorder the terms:
0.64 + 1.6n + n2 = 2 + 0.64

Combine like terms: 2 + 0.64 = 2.64
0.64 + 1.6n + n2 = 2.64

Factor a perfect square on the left side:
(n + 0.8)(n + 0.8) = 2.64

Calculate the square root of the right side: 1.624807681

Break this problem into two subproblems by setting 
(n + 0.8) equal to 1.624807681 and -1.624807681.

Subproblem 1

n + 0.8 = 1.624807681 Simplifying n + 0.8 = 1.624807681 Reorder the terms: 0.8 + n = 1.624807681 Solving 0.8 + n = 1.624807681 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.8' to each side of the equation. 0.8 + -0.8 + n = 1.624807681 + -0.8 Combine like terms: 0.8 + -0.8 = 0.0 0.0 + n = 1.624807681 + -0.8 n = 1.624807681 + -0.8 Combine like terms: 1.624807681 + -0.8 = 0.824807681 n = 0.824807681 Simplifying n = 0.824807681

Subproblem 2

n + 0.8 = -1.624807681 Simplifying n + 0.8 = -1.624807681 Reorder the terms: 0.8 + n = -1.624807681 Solving 0.8 + n = -1.624807681 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.8' to each side of the equation. 0.8 + -0.8 + n = -1.624807681 + -0.8 Combine like terms: 0.8 + -0.8 = 0.0 0.0 + n = -1.624807681 + -0.8 n = -1.624807681 + -0.8 Combine like terms: -1.624807681 + -0.8 = -2.424807681 n = -2.424807681 Simplifying n = -2.424807681

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.824807681, -2.424807681}

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