16n^2-40n=-23

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Solution for 16n^2-40n=-23 equation:


Simplifying
16n2 + -40n = -23

Reorder the terms:
-40n + 16n2 = -23

Solving
-40n + 16n2 = -23

Solving for variable 'n'.

Reorder the terms:
23 + -40n + 16n2 = -23 + 23

Combine like terms: -23 + 23 = 0
23 + -40n + 16n2 = 0

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

Divide each side by '16'.
1.4375 + -2.5n + n2 = 0

Move the constant term to the right:

Add '-1.4375' to each side of the equation.
1.4375 + -2.5n + -1.4375 + n2 = 0 + -1.4375

Reorder the terms:
1.4375 + -1.4375 + -2.5n + n2 = 0 + -1.4375

Combine like terms: 1.4375 + -1.4375 = 0.0000
0.0000 + -2.5n + n2 = 0 + -1.4375
-2.5n + n2 = 0 + -1.4375

Combine like terms: 0 + -1.4375 = -1.4375
-2.5n + n2 = -1.4375

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 = -1.4375 + 1.5625

Reorder the terms:
1.5625 + -2.5n + n2 = -1.4375 + 1.5625

Combine like terms: -1.4375 + 1.5625 = 0.125
1.5625 + -2.5n + n2 = 0.125

Factor a perfect square on the left side:
(n + -1.25)(n + -1.25) = 0.125

Calculate the square root of the right side: 0.353553391

Break this problem into two subproblems by setting 
(n + -1.25) equal to 0.353553391 and -0.353553391.

Subproblem 1

n + -1.25 = 0.353553391 Simplifying n + -1.25 = 0.353553391 Reorder the terms: -1.25 + n = 0.353553391 Solving -1.25 + n = 0.353553391 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 = 0.353553391 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + n = 0.353553391 + 1.25 n = 0.353553391 + 1.25 Combine like terms: 0.353553391 + 1.25 = 1.603553391 n = 1.603553391 Simplifying n = 1.603553391

Subproblem 2

n + -1.25 = -0.353553391 Simplifying n + -1.25 = -0.353553391 Reorder the terms: -1.25 + n = -0.353553391 Solving -1.25 + n = -0.353553391 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 = -0.353553391 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + n = -0.353553391 + 1.25 n = -0.353553391 + 1.25 Combine like terms: -0.353553391 + 1.25 = 0.896446609 n = 0.896446609 Simplifying n = 0.896446609

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.603553391, 0.896446609}

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