5n^2+1=-8n

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Solution for 5n^2+1=-8n equation:


Simplifying
5n2 + 1 = -8n

Reorder the terms:
1 + 5n2 = -8n

Solving
1 + 5n2 = -8n

Solving for variable 'n'.

Reorder the terms:
1 + 8n + 5n2 = -8n + 8n

Combine like terms: -8n + 8n = 0
1 + 8n + 5n2 = 0

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

Divide each side by '5'.
0.2 + 1.6n + n2 = 0

Move the constant term to the right:

Add '-0.2' to each side of the equation.
0.2 + 1.6n + -0.2 + n2 = 0 + -0.2

Reorder the terms:
0.2 + -0.2 + 1.6n + n2 = 0 + -0.2

Combine like terms: 0.2 + -0.2 = 0.0
0.0 + 1.6n + n2 = 0 + -0.2
1.6n + n2 = 0 + -0.2

Combine like terms: 0 + -0.2 = -0.2
1.6n + n2 = -0.2

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 = -0.2 + 0.64

Reorder the terms:
0.64 + 1.6n + n2 = -0.2 + 0.64

Combine like terms: -0.2 + 0.64 = 0.44
0.64 + 1.6n + n2 = 0.44

Factor a perfect square on the left side:
(n + 0.8)(n + 0.8) = 0.44

Calculate the square root of the right side: 0.663324958

Break this problem into two subproblems by setting 
(n + 0.8) equal to 0.663324958 and -0.663324958.

Subproblem 1

n + 0.8 = 0.663324958 Simplifying n + 0.8 = 0.663324958 Reorder the terms: 0.8 + n = 0.663324958 Solving 0.8 + n = 0.663324958 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 = 0.663324958 + -0.8 Combine like terms: 0.8 + -0.8 = 0.0 0.0 + n = 0.663324958 + -0.8 n = 0.663324958 + -0.8 Combine like terms: 0.663324958 + -0.8 = -0.136675042 n = -0.136675042 Simplifying n = -0.136675042

Subproblem 2

n + 0.8 = -0.663324958 Simplifying n + 0.8 = -0.663324958 Reorder the terms: 0.8 + n = -0.663324958 Solving 0.8 + n = -0.663324958 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 = -0.663324958 + -0.8 Combine like terms: 0.8 + -0.8 = 0.0 0.0 + n = -0.663324958 + -0.8 n = -0.663324958 + -0.8 Combine like terms: -0.663324958 + -0.8 = -1.463324958 n = -1.463324958 Simplifying n = -1.463324958

Solution

The solution to the problem is based on the solutions from the subproblems. n = {-0.136675042, -1.463324958}

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