8n(4n+7)=13

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Solution for 8n(4n+7)=13 equation:


Simplifying
8n(4n + 7) = 13

Reorder the terms:
8n(7 + 4n) = 13
(7 * 8n + 4n * 8n) = 13
(56n + 32n2) = 13

Solving
56n + 32n2 = 13

Solving for variable 'n'.

Reorder the terms:
-13 + 56n + 32n2 = 13 + -13

Combine like terms: 13 + -13 = 0
-13 + 56n + 32n2 = 0

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

Divide each side by '32'.
-0.40625 + 1.75n + n2 = 0

Move the constant term to the right:

Add '0.40625' to each side of the equation.
-0.40625 + 1.75n + 0.40625 + n2 = 0 + 0.40625

Reorder the terms:
-0.40625 + 0.40625 + 1.75n + n2 = 0 + 0.40625

Combine like terms: -0.40625 + 0.40625 = 0.00000
0.00000 + 1.75n + n2 = 0 + 0.40625
1.75n + n2 = 0 + 0.40625

Combine like terms: 0 + 0.40625 = 0.40625
1.75n + n2 = 0.40625

The n term is 1.75n.  Take half its coefficient (0.875).
Square it (0.765625) and add it to both sides.

Add '0.765625' to each side of the equation.
1.75n + 0.765625 + n2 = 0.40625 + 0.765625

Reorder the terms:
0.765625 + 1.75n + n2 = 0.40625 + 0.765625

Combine like terms: 0.40625 + 0.765625 = 1.171875
0.765625 + 1.75n + n2 = 1.171875

Factor a perfect square on the left side:
(n + 0.875)(n + 0.875) = 1.171875

Calculate the square root of the right side: 1.082531755

Break this problem into two subproblems by setting 
(n + 0.875) equal to 1.082531755 and -1.082531755.

Subproblem 1

n + 0.875 = 1.082531755 Simplifying n + 0.875 = 1.082531755 Reorder the terms: 0.875 + n = 1.082531755 Solving 0.875 + n = 1.082531755 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.875' to each side of the equation. 0.875 + -0.875 + n = 1.082531755 + -0.875 Combine like terms: 0.875 + -0.875 = 0.000 0.000 + n = 1.082531755 + -0.875 n = 1.082531755 + -0.875 Combine like terms: 1.082531755 + -0.875 = 0.207531755 n = 0.207531755 Simplifying n = 0.207531755

Subproblem 2

n + 0.875 = -1.082531755 Simplifying n + 0.875 = -1.082531755 Reorder the terms: 0.875 + n = -1.082531755 Solving 0.875 + n = -1.082531755 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.875' to each side of the equation. 0.875 + -0.875 + n = -1.082531755 + -0.875 Combine like terms: 0.875 + -0.875 = 0.000 0.000 + n = -1.082531755 + -0.875 n = -1.082531755 + -0.875 Combine like terms: -1.082531755 + -0.875 = -1.957531755 n = -1.957531755 Simplifying n = -1.957531755

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.207531755, -1.957531755}

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