21n^2-38n=38

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Solution for 21n^2-38n=38 equation:


Simplifying
21n2 + -38n = 38

Reorder the terms:
-38n + 21n2 = 38

Solving
-38n + 21n2 = 38

Solving for variable 'n'.

Reorder the terms:
-38 + -38n + 21n2 = 38 + -38

Combine like terms: 38 + -38 = 0
-38 + -38n + 21n2 = 0

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

Divide each side by '21'.
-1.80952381 + -1.80952381n + n2 = 0

Move the constant term to the right:

Add '1.80952381' to each side of the equation.
-1.80952381 + -1.80952381n + 1.80952381 + n2 = 0 + 1.80952381

Reorder the terms:
-1.80952381 + 1.80952381 + -1.80952381n + n2 = 0 + 1.80952381

Combine like terms: -1.80952381 + 1.80952381 = 0.00000000
0.00000000 + -1.80952381n + n2 = 0 + 1.80952381
-1.80952381n + n2 = 0 + 1.80952381

Combine like terms: 0 + 1.80952381 = 1.80952381
-1.80952381n + n2 = 1.80952381

The n term is -1.80952381n.  Take half its coefficient (-0.904761905).
Square it (0.8185941047) and add it to both sides.

Add '0.8185941047' to each side of the equation.
-1.80952381n + 0.8185941047 + n2 = 1.80952381 + 0.8185941047

Reorder the terms:
0.8185941047 + -1.80952381n + n2 = 1.80952381 + 0.8185941047

Combine like terms: 1.80952381 + 0.8185941047 = 2.6281179147
0.8185941047 + -1.80952381n + n2 = 2.6281179147

Factor a perfect square on the left side:
(n + -0.904761905)(n + -0.904761905) = 2.6281179147

Calculate the square root of the right side: 1.621147098

Break this problem into two subproblems by setting 
(n + -0.904761905) equal to 1.621147098 and -1.621147098.

Subproblem 1

n + -0.904761905 = 1.621147098 Simplifying n + -0.904761905 = 1.621147098 Reorder the terms: -0.904761905 + n = 1.621147098 Solving -0.904761905 + n = 1.621147098 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.904761905' to each side of the equation. -0.904761905 + 0.904761905 + n = 1.621147098 + 0.904761905 Combine like terms: -0.904761905 + 0.904761905 = 0.000000000 0.000000000 + n = 1.621147098 + 0.904761905 n = 1.621147098 + 0.904761905 Combine like terms: 1.621147098 + 0.904761905 = 2.525909003 n = 2.525909003 Simplifying n = 2.525909003

Subproblem 2

n + -0.904761905 = -1.621147098 Simplifying n + -0.904761905 = -1.621147098 Reorder the terms: -0.904761905 + n = -1.621147098 Solving -0.904761905 + n = -1.621147098 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.904761905' to each side of the equation. -0.904761905 + 0.904761905 + n = -1.621147098 + 0.904761905 Combine like terms: -0.904761905 + 0.904761905 = 0.000000000 0.000000000 + n = -1.621147098 + 0.904761905 n = -1.621147098 + 0.904761905 Combine like terms: -1.621147098 + 0.904761905 = -0.716385193 n = -0.716385193 Simplifying n = -0.716385193

Solution

The solution to the problem is based on the solutions from the subproblems. n = {2.525909003, -0.716385193}

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