8n^2-23=7n

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Solution for 8n^2-23=7n equation:


Simplifying
8n2 + -23 = 7n

Reorder the terms:
-23 + 8n2 = 7n

Solving
-23 + 8n2 = 7n

Solving for variable 'n'.

Reorder the terms:
-23 + -7n + 8n2 = 7n + -7n

Combine like terms: 7n + -7n = 0
-23 + -7n + 8n2 = 0

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

Divide each side by '8'.
-2.875 + -0.875n + n2 = 0

Move the constant term to the right:

Add '2.875' to each side of the equation.
-2.875 + -0.875n + 2.875 + n2 = 0 + 2.875

Reorder the terms:
-2.875 + 2.875 + -0.875n + n2 = 0 + 2.875

Combine like terms: -2.875 + 2.875 = 0.000
0.000 + -0.875n + n2 = 0 + 2.875
-0.875n + n2 = 0 + 2.875

Combine like terms: 0 + 2.875 = 2.875
-0.875n + n2 = 2.875

The n term is -0.875n.  Take half its coefficient (-0.4375).
Square it (0.19140625) and add it to both sides.

Add '0.19140625' to each side of the equation.
-0.875n + 0.19140625 + n2 = 2.875 + 0.19140625

Reorder the terms:
0.19140625 + -0.875n + n2 = 2.875 + 0.19140625

Combine like terms: 2.875 + 0.19140625 = 3.06640625
0.19140625 + -0.875n + n2 = 3.06640625

Factor a perfect square on the left side:
(n + -0.4375)(n + -0.4375) = 3.06640625

Calculate the square root of the right side: 1.751115716

Break this problem into two subproblems by setting 
(n + -0.4375) equal to 1.751115716 and -1.751115716.

Subproblem 1

n + -0.4375 = 1.751115716 Simplifying n + -0.4375 = 1.751115716 Reorder the terms: -0.4375 + n = 1.751115716 Solving -0.4375 + n = 1.751115716 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.4375' to each side of the equation. -0.4375 + 0.4375 + n = 1.751115716 + 0.4375 Combine like terms: -0.4375 + 0.4375 = 0.0000 0.0000 + n = 1.751115716 + 0.4375 n = 1.751115716 + 0.4375 Combine like terms: 1.751115716 + 0.4375 = 2.188615716 n = 2.188615716 Simplifying n = 2.188615716

Subproblem 2

n + -0.4375 = -1.751115716 Simplifying n + -0.4375 = -1.751115716 Reorder the terms: -0.4375 + n = -1.751115716 Solving -0.4375 + n = -1.751115716 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.4375' to each side of the equation. -0.4375 + 0.4375 + n = -1.751115716 + 0.4375 Combine like terms: -0.4375 + 0.4375 = 0.0000 0.0000 + n = -1.751115716 + 0.4375 n = -1.751115716 + 0.4375 Combine like terms: -1.751115716 + 0.4375 = -1.313615716 n = -1.313615716 Simplifying n = -1.313615716

Solution

The solution to the problem is based on the solutions from the subproblems. n = {2.188615716, -1.313615716}

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