8n^2+9n=10

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Solution for 8n^2+9n=10 equation:


Simplifying
8n2 + 9n = 10

Reorder the terms:
9n + 8n2 = 10

Solving
9n + 8n2 = 10

Solving for variable 'n'.

Reorder the terms:
-10 + 9n + 8n2 = 10 + -10

Combine like terms: 10 + -10 = 0
-10 + 9n + 8n2 = 0

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

Divide each side by '8'.
-1.25 + 1.125n + n2 = 0

Move the constant term to the right:

Add '1.25' to each side of the equation.
-1.25 + 1.125n + 1.25 + n2 = 0 + 1.25

Reorder the terms:
-1.25 + 1.25 + 1.125n + n2 = 0 + 1.25

Combine like terms: -1.25 + 1.25 = 0.00
0.00 + 1.125n + n2 = 0 + 1.25
1.125n + n2 = 0 + 1.25

Combine like terms: 0 + 1.25 = 1.25
1.125n + n2 = 1.25

The n term is 1.125n.  Take half its coefficient (0.5625).
Square it (0.31640625) and add it to both sides.

Add '0.31640625' to each side of the equation.
1.125n + 0.31640625 + n2 = 1.25 + 0.31640625

Reorder the terms:
0.31640625 + 1.125n + n2 = 1.25 + 0.31640625

Combine like terms: 1.25 + 0.31640625 = 1.56640625
0.31640625 + 1.125n + n2 = 1.56640625

Factor a perfect square on the left side:
(n + 0.5625)(n + 0.5625) = 1.56640625

Calculate the square root of the right side: 1.251561525

Break this problem into two subproblems by setting 
(n + 0.5625) equal to 1.251561525 and -1.251561525.

Subproblem 1

n + 0.5625 = 1.251561525 Simplifying n + 0.5625 = 1.251561525 Reorder the terms: 0.5625 + n = 1.251561525 Solving 0.5625 + n = 1.251561525 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5625' to each side of the equation. 0.5625 + -0.5625 + n = 1.251561525 + -0.5625 Combine like terms: 0.5625 + -0.5625 = 0.0000 0.0000 + n = 1.251561525 + -0.5625 n = 1.251561525 + -0.5625 Combine like terms: 1.251561525 + -0.5625 = 0.689061525 n = 0.689061525 Simplifying n = 0.689061525

Subproblem 2

n + 0.5625 = -1.251561525 Simplifying n + 0.5625 = -1.251561525 Reorder the terms: 0.5625 + n = -1.251561525 Solving 0.5625 + n = -1.251561525 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5625' to each side of the equation. 0.5625 + -0.5625 + n = -1.251561525 + -0.5625 Combine like terms: 0.5625 + -0.5625 = 0.0000 0.0000 + n = -1.251561525 + -0.5625 n = -1.251561525 + -0.5625 Combine like terms: -1.251561525 + -0.5625 = -1.814061525 n = -1.814061525 Simplifying n = -1.814061525

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.689061525, -1.814061525}

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