2n^2-21n+7=0

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Solution for 2n^2-21n+7=0 equation:


Simplifying
2n2 + -21n + 7 = 0

Reorder the terms:
7 + -21n + 2n2 = 0

Solving
7 + -21n + 2n2 = 0

Solving for variable 'n'.

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

Divide each side by '2'.
3.5 + -10.5n + n2 = 0

Move the constant term to the right:

Add '-3.5' to each side of the equation.
3.5 + -10.5n + -3.5 + n2 = 0 + -3.5

Reorder the terms:
3.5 + -3.5 + -10.5n + n2 = 0 + -3.5

Combine like terms: 3.5 + -3.5 = 0.0
0.0 + -10.5n + n2 = 0 + -3.5
-10.5n + n2 = 0 + -3.5

Combine like terms: 0 + -3.5 = -3.5
-10.5n + n2 = -3.5

The n term is -10.5n.  Take half its coefficient (-5.25).
Square it (27.5625) and add it to both sides.

Add '27.5625' to each side of the equation.
-10.5n + 27.5625 + n2 = -3.5 + 27.5625

Reorder the terms:
27.5625 + -10.5n + n2 = -3.5 + 27.5625

Combine like terms: -3.5 + 27.5625 = 24.0625
27.5625 + -10.5n + n2 = 24.0625

Factor a perfect square on the left side:
(n + -5.25)(n + -5.25) = 24.0625

Calculate the square root of the right side: 4.905354218

Break this problem into two subproblems by setting 
(n + -5.25) equal to 4.905354218 and -4.905354218.

Subproblem 1

n + -5.25 = 4.905354218 Simplifying n + -5.25 = 4.905354218 Reorder the terms: -5.25 + n = 4.905354218 Solving -5.25 + n = 4.905354218 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '5.25' to each side of the equation. -5.25 + 5.25 + n = 4.905354218 + 5.25 Combine like terms: -5.25 + 5.25 = 0.00 0.00 + n = 4.905354218 + 5.25 n = 4.905354218 + 5.25 Combine like terms: 4.905354218 + 5.25 = 10.155354218 n = 10.155354218 Simplifying n = 10.155354218

Subproblem 2

n + -5.25 = -4.905354218 Simplifying n + -5.25 = -4.905354218 Reorder the terms: -5.25 + n = -4.905354218 Solving -5.25 + n = -4.905354218 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '5.25' to each side of the equation. -5.25 + 5.25 + n = -4.905354218 + 5.25 Combine like terms: -5.25 + 5.25 = 0.00 0.00 + n = -4.905354218 + 5.25 n = -4.905354218 + 5.25 Combine like terms: -4.905354218 + 5.25 = 0.344645782 n = 0.344645782 Simplifying n = 0.344645782

Solution

The solution to the problem is based on the solutions from the subproblems. n = {10.155354218, 0.344645782}

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