16n^2+8n-25=0

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Solution for 16n^2+8n-25=0 equation:


Simplifying
16n2 + 8n + -25 = 0

Reorder the terms:
-25 + 8n + 16n2 = 0

Solving
-25 + 8n + 16n2 = 0

Solving for variable 'n'.

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

Divide each side by '16'.
-1.5625 + 0.5n + n2 = 0

Move the constant term to the right:

Add '1.5625' to each side of the equation.
-1.5625 + 0.5n + 1.5625 + n2 = 0 + 1.5625

Reorder the terms:
-1.5625 + 1.5625 + 0.5n + n2 = 0 + 1.5625

Combine like terms: -1.5625 + 1.5625 = 0.0000
0.0000 + 0.5n + n2 = 0 + 1.5625
0.5n + n2 = 0 + 1.5625

Combine like terms: 0 + 1.5625 = 1.5625
0.5n + n2 = 1.5625

The n term is 0.5n.  Take half its coefficient (0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
0.5n + 0.0625 + n2 = 1.5625 + 0.0625

Reorder the terms:
0.0625 + 0.5n + n2 = 1.5625 + 0.0625

Combine like terms: 1.5625 + 0.0625 = 1.625
0.0625 + 0.5n + n2 = 1.625

Factor a perfect square on the left side:
(n + 0.25)(n + 0.25) = 1.625

Calculate the square root of the right side: 1.274754878

Break this problem into two subproblems by setting 
(n + 0.25) equal to 1.274754878 and -1.274754878.

Subproblem 1

n + 0.25 = 1.274754878 Simplifying n + 0.25 = 1.274754878 Reorder the terms: 0.25 + n = 1.274754878 Solving 0.25 + n = 1.274754878 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + n = 1.274754878 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + n = 1.274754878 + -0.25 n = 1.274754878 + -0.25 Combine like terms: 1.274754878 + -0.25 = 1.024754878 n = 1.024754878 Simplifying n = 1.024754878

Subproblem 2

n + 0.25 = -1.274754878 Simplifying n + 0.25 = -1.274754878 Reorder the terms: 0.25 + n = -1.274754878 Solving 0.25 + n = -1.274754878 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + n = -1.274754878 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + n = -1.274754878 + -0.25 n = -1.274754878 + -0.25 Combine like terms: -1.274754878 + -0.25 = -1.524754878 n = -1.524754878 Simplifying n = -1.524754878

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.024754878, -1.524754878}

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