16n^2-56n=-47

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Solution for 16n^2-56n=-47 equation:


Simplifying
16n2 + -56n = -47

Reorder the terms:
-56n + 16n2 = -47

Solving
-56n + 16n2 = -47

Solving for variable 'n'.

Reorder the terms:
47 + -56n + 16n2 = -47 + 47

Combine like terms: -47 + 47 = 0
47 + -56n + 16n2 = 0

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

Divide each side by '16'.
2.9375 + -3.5n + n2 = 0

Move the constant term to the right:

Add '-2.9375' to each side of the equation.
2.9375 + -3.5n + -2.9375 + n2 = 0 + -2.9375

Reorder the terms:
2.9375 + -2.9375 + -3.5n + n2 = 0 + -2.9375

Combine like terms: 2.9375 + -2.9375 = 0.0000
0.0000 + -3.5n + n2 = 0 + -2.9375
-3.5n + n2 = 0 + -2.9375

Combine like terms: 0 + -2.9375 = -2.9375
-3.5n + n2 = -2.9375

The n term is -3.5n.  Take half its coefficient (-1.75).
Square it (3.0625) and add it to both sides.

Add '3.0625' to each side of the equation.
-3.5n + 3.0625 + n2 = -2.9375 + 3.0625

Reorder the terms:
3.0625 + -3.5n + n2 = -2.9375 + 3.0625

Combine like terms: -2.9375 + 3.0625 = 0.125
3.0625 + -3.5n + n2 = 0.125

Factor a perfect square on the left side:
(n + -1.75)(n + -1.75) = 0.125

Calculate the square root of the right side: 0.353553391

Break this problem into two subproblems by setting 
(n + -1.75) equal to 0.353553391 and -0.353553391.

Subproblem 1

n + -1.75 = 0.353553391 Simplifying n + -1.75 = 0.353553391 Reorder the terms: -1.75 + n = 0.353553391 Solving -1.75 + n = 0.353553391 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.75' to each side of the equation. -1.75 + 1.75 + n = 0.353553391 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + n = 0.353553391 + 1.75 n = 0.353553391 + 1.75 Combine like terms: 0.353553391 + 1.75 = 2.103553391 n = 2.103553391 Simplifying n = 2.103553391

Subproblem 2

n + -1.75 = -0.353553391 Simplifying n + -1.75 = -0.353553391 Reorder the terms: -1.75 + n = -0.353553391 Solving -1.75 + n = -0.353553391 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.75' to each side of the equation. -1.75 + 1.75 + n = -0.353553391 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + n = -0.353553391 + 1.75 n = -0.353553391 + 1.75 Combine like terms: -0.353553391 + 1.75 = 1.396446609 n = 1.396446609 Simplifying n = 1.396446609

Solution

The solution to the problem is based on the solutions from the subproblems. n = {2.103553391, 1.396446609}

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