8n^2-3n=10

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Solution for 8n^2-3n=10 equation:


Simplifying
8n2 + -3n = 10

Reorder the terms:
-3n + 8n2 = 10

Solving
-3n + 8n2 = 10

Solving for variable 'n'.

Reorder the terms:
-10 + -3n + 8n2 = 10 + -10

Combine like terms: 10 + -10 = 0
-10 + -3n + 8n2 = 0

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

Divide each side by '8'.
-1.25 + -0.375n + n2 = 0

Move the constant term to the right:

Add '1.25' to each side of the equation.
-1.25 + -0.375n + 1.25 + n2 = 0 + 1.25

Reorder the terms:
-1.25 + 1.25 + -0.375n + n2 = 0 + 1.25

Combine like terms: -1.25 + 1.25 = 0.00
0.00 + -0.375n + n2 = 0 + 1.25
-0.375n + n2 = 0 + 1.25

Combine like terms: 0 + 1.25 = 1.25
-0.375n + n2 = 1.25

The n term is -0.375n.  Take half its coefficient (-0.1875).
Square it (0.03515625) and add it to both sides.

Add '0.03515625' to each side of the equation.
-0.375n + 0.03515625 + n2 = 1.25 + 0.03515625

Reorder the terms:
0.03515625 + -0.375n + n2 = 1.25 + 0.03515625

Combine like terms: 1.25 + 0.03515625 = 1.28515625
0.03515625 + -0.375n + n2 = 1.28515625

Factor a perfect square on the left side:
(n + -0.1875)(n + -0.1875) = 1.28515625

Calculate the square root of the right side: 1.133647322

Break this problem into two subproblems by setting 
(n + -0.1875) equal to 1.133647322 and -1.133647322.

Subproblem 1

n + -0.1875 = 1.133647322 Simplifying n + -0.1875 = 1.133647322 Reorder the terms: -0.1875 + n = 1.133647322 Solving -0.1875 + n = 1.133647322 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.1875' to each side of the equation. -0.1875 + 0.1875 + n = 1.133647322 + 0.1875 Combine like terms: -0.1875 + 0.1875 = 0.0000 0.0000 + n = 1.133647322 + 0.1875 n = 1.133647322 + 0.1875 Combine like terms: 1.133647322 + 0.1875 = 1.321147322 n = 1.321147322 Simplifying n = 1.321147322

Subproblem 2

n + -0.1875 = -1.133647322 Simplifying n + -0.1875 = -1.133647322 Reorder the terms: -0.1875 + n = -1.133647322 Solving -0.1875 + n = -1.133647322 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.1875' to each side of the equation. -0.1875 + 0.1875 + n = -1.133647322 + 0.1875 Combine like terms: -0.1875 + 0.1875 = 0.0000 0.0000 + n = -1.133647322 + 0.1875 n = -1.133647322 + 0.1875 Combine like terms: -1.133647322 + 0.1875 = -0.946147322 n = -0.946147322 Simplifying n = -0.946147322

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.321147322, -0.946147322}

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