8n(2n+5)=25

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Solution for 8n(2n+5)=25 equation:


Simplifying
8n(2n + 5) = 25

Reorder the terms:
8n(5 + 2n) = 25
(5 * 8n + 2n * 8n) = 25
(40n + 16n2) = 25

Solving
40n + 16n2 = 25

Solving for variable 'n'.

Reorder the terms:
-25 + 40n + 16n2 = 25 + -25

Combine like terms: 25 + -25 = 0
-25 + 40n + 16n2 = 0

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

Divide each side by '16'.
-1.5625 + 2.5n + n2 = 0

Move the constant term to the right:

Add '1.5625' to each side of the equation.
-1.5625 + 2.5n + 1.5625 + n2 = 0 + 1.5625

Reorder the terms:
-1.5625 + 1.5625 + 2.5n + n2 = 0 + 1.5625

Combine like terms: -1.5625 + 1.5625 = 0.0000
0.0000 + 2.5n + n2 = 0 + 1.5625
2.5n + n2 = 0 + 1.5625

Combine like terms: 0 + 1.5625 = 1.5625
2.5n + n2 = 1.5625

The n term is 2.5n.  Take half its coefficient (1.25).
Square it (1.5625) and add it to both sides.

Add '1.5625' to each side of the equation.
2.5n + 1.5625 + n2 = 1.5625 + 1.5625

Reorder the terms:
1.5625 + 2.5n + n2 = 1.5625 + 1.5625

Combine like terms: 1.5625 + 1.5625 = 3.125
1.5625 + 2.5n + n2 = 3.125

Factor a perfect square on the left side:
(n + 1.25)(n + 1.25) = 3.125

Calculate the square root of the right side: 1.767766953

Break this problem into two subproblems by setting 
(n + 1.25) equal to 1.767766953 and -1.767766953.

Subproblem 1

n + 1.25 = 1.767766953 Simplifying n + 1.25 = 1.767766953 Reorder the terms: 1.25 + n = 1.767766953 Solving 1.25 + n = 1.767766953 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + n = 1.767766953 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + n = 1.767766953 + -1.25 n = 1.767766953 + -1.25 Combine like terms: 1.767766953 + -1.25 = 0.517766953 n = 0.517766953 Simplifying n = 0.517766953

Subproblem 2

n + 1.25 = -1.767766953 Simplifying n + 1.25 = -1.767766953 Reorder the terms: 1.25 + n = -1.767766953 Solving 1.25 + n = -1.767766953 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + n = -1.767766953 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + n = -1.767766953 + -1.25 n = -1.767766953 + -1.25 Combine like terms: -1.767766953 + -1.25 = -3.017766953 n = -3.017766953 Simplifying n = -3.017766953

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.517766953, -3.017766953}

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