10n^2=8n+7

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Solution for 10n^2=8n+7 equation:


Simplifying
10n2 = 8n + 7

Reorder the terms:
10n2 = 7 + 8n

Solving
10n2 = 7 + 8n

Solving for variable 'n'.

Reorder the terms:
-7 + -8n + 10n2 = 7 + 8n + -7 + -8n

Reorder the terms:
-7 + -8n + 10n2 = 7 + -7 + 8n + -8n

Combine like terms: 7 + -7 = 0
-7 + -8n + 10n2 = 0 + 8n + -8n
-7 + -8n + 10n2 = 8n + -8n

Combine like terms: 8n + -8n = 0
-7 + -8n + 10n2 = 0

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

Divide each side by '10'.
-0.7 + -0.8n + n2 = 0

Move the constant term to the right:

Add '0.7' to each side of the equation.
-0.7 + -0.8n + 0.7 + n2 = 0 + 0.7

Reorder the terms:
-0.7 + 0.7 + -0.8n + n2 = 0 + 0.7

Combine like terms: -0.7 + 0.7 = 0.0
0.0 + -0.8n + n2 = 0 + 0.7
-0.8n + n2 = 0 + 0.7

Combine like terms: 0 + 0.7 = 0.7
-0.8n + n2 = 0.7

The n term is -0.8n.  Take half its coefficient (-0.4).
Square it (0.16) and add it to both sides.

Add '0.16' to each side of the equation.
-0.8n + 0.16 + n2 = 0.7 + 0.16

Reorder the terms:
0.16 + -0.8n + n2 = 0.7 + 0.16

Combine like terms: 0.7 + 0.16 = 0.86
0.16 + -0.8n + n2 = 0.86

Factor a perfect square on the left side:
(n + -0.4)(n + -0.4) = 0.86

Calculate the square root of the right side: 0.92736185

Break this problem into two subproblems by setting 
(n + -0.4) equal to 0.92736185 and -0.92736185.

Subproblem 1

n + -0.4 = 0.92736185 Simplifying n + -0.4 = 0.92736185 Reorder the terms: -0.4 + n = 0.92736185 Solving -0.4 + n = 0.92736185 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.4' to each side of the equation. -0.4 + 0.4 + n = 0.92736185 + 0.4 Combine like terms: -0.4 + 0.4 = 0.0 0.0 + n = 0.92736185 + 0.4 n = 0.92736185 + 0.4 Combine like terms: 0.92736185 + 0.4 = 1.32736185 n = 1.32736185 Simplifying n = 1.32736185

Subproblem 2

n + -0.4 = -0.92736185 Simplifying n + -0.4 = -0.92736185 Reorder the terms: -0.4 + n = -0.92736185 Solving -0.4 + n = -0.92736185 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.4' to each side of the equation. -0.4 + 0.4 + n = -0.92736185 + 0.4 Combine like terms: -0.4 + 0.4 = 0.0 0.0 + n = -0.92736185 + 0.4 n = -0.92736185 + 0.4 Combine like terms: -0.92736185 + 0.4 = -0.52736185 n = -0.52736185 Simplifying n = -0.52736185

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.32736185, -0.52736185}

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