8n(4n+7)=25

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Solution for 8n(4n+7)=25 equation:


Simplifying
8n(4n + 7) = 25

Reorder the terms:
8n(7 + 4n) = 25
(7 * 8n + 4n * 8n) = 25
(56n + 32n2) = 25

Solving
56n + 32n2 = 25

Solving for variable 'n'.

Reorder the terms:
-25 + 56n + 32n2 = 25 + -25

Combine like terms: 25 + -25 = 0
-25 + 56n + 32n2 = 0

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

Divide each side by '32'.
-0.78125 + 1.75n + n2 = 0

Move the constant term to the right:

Add '0.78125' to each side of the equation.
-0.78125 + 1.75n + 0.78125 + n2 = 0 + 0.78125

Reorder the terms:
-0.78125 + 0.78125 + 1.75n + n2 = 0 + 0.78125

Combine like terms: -0.78125 + 0.78125 = 0.00000
0.00000 + 1.75n + n2 = 0 + 0.78125
1.75n + n2 = 0 + 0.78125

Combine like terms: 0 + 0.78125 = 0.78125
1.75n + n2 = 0.78125

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 = 0.78125 + 0.765625

Reorder the terms:
0.765625 + 1.75n + n2 = 0.78125 + 0.765625

Combine like terms: 0.78125 + 0.765625 = 1.546875
0.765625 + 1.75n + n2 = 1.546875

Factor a perfect square on the left side:
(n + 0.875)(n + 0.875) = 1.546875

Calculate the square root of the right side: 1.243734296

Break this problem into two subproblems by setting 
(n + 0.875) equal to 1.243734296 and -1.243734296.

Subproblem 1

n + 0.875 = 1.243734296 Simplifying n + 0.875 = 1.243734296 Reorder the terms: 0.875 + n = 1.243734296 Solving 0.875 + n = 1.243734296 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 = 1.243734296 + -0.875 Combine like terms: 0.875 + -0.875 = 0.000 0.000 + n = 1.243734296 + -0.875 n = 1.243734296 + -0.875 Combine like terms: 1.243734296 + -0.875 = 0.368734296 n = 0.368734296 Simplifying n = 0.368734296

Subproblem 2

n + 0.875 = -1.243734296 Simplifying n + 0.875 = -1.243734296 Reorder the terms: 0.875 + n = -1.243734296 Solving 0.875 + n = -1.243734296 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 = -1.243734296 + -0.875 Combine like terms: 0.875 + -0.875 = 0.000 0.000 + n = -1.243734296 + -0.875 n = -1.243734296 + -0.875 Combine like terms: -1.243734296 + -0.875 = -2.118734296 n = -2.118734296 Simplifying n = -2.118734296

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.368734296, -2.118734296}

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