7n^2-32n+20=0

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Solution for 7n^2-32n+20=0 equation:


Simplifying
7n2 + -32n + 20 = 0

Reorder the terms:
20 + -32n + 7n2 = 0

Solving
20 + -32n + 7n2 = 0

Solving for variable 'n'.

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

Divide each side by '7'.
2.857142857 + -4.571428571n + n2 = 0

Move the constant term to the right:

Add '-2.857142857' to each side of the equation.
2.857142857 + -4.571428571n + -2.857142857 + n2 = 0 + -2.857142857

Reorder the terms:
2.857142857 + -2.857142857 + -4.571428571n + n2 = 0 + -2.857142857

Combine like terms: 2.857142857 + -2.857142857 = 0.000000000
0.000000000 + -4.571428571n + n2 = 0 + -2.857142857
-4.571428571n + n2 = 0 + -2.857142857

Combine like terms: 0 + -2.857142857 = -2.857142857
-4.571428571n + n2 = -2.857142857

The n term is -4.571428571n.  Take half its coefficient (-2.285714286).
Square it (5.224489797) and add it to both sides.

Add '5.224489797' to each side of the equation.
-4.571428571n + 5.224489797 + n2 = -2.857142857 + 5.224489797

Reorder the terms:
5.224489797 + -4.571428571n + n2 = -2.857142857 + 5.224489797

Combine like terms: -2.857142857 + 5.224489797 = 2.36734694
5.224489797 + -4.571428571n + n2 = 2.36734694

Factor a perfect square on the left side:
(n + -2.285714286)(n + -2.285714286) = 2.36734694

Calculate the square root of the right side: 1.538618517

Break this problem into two subproblems by setting 
(n + -2.285714286) equal to 1.538618517 and -1.538618517.

Subproblem 1

n + -2.285714286 = 1.538618517 Simplifying n + -2.285714286 = 1.538618517 Reorder the terms: -2.285714286 + n = 1.538618517 Solving -2.285714286 + n = 1.538618517 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2.285714286' to each side of the equation. -2.285714286 + 2.285714286 + n = 1.538618517 + 2.285714286 Combine like terms: -2.285714286 + 2.285714286 = 0.000000000 0.000000000 + n = 1.538618517 + 2.285714286 n = 1.538618517 + 2.285714286 Combine like terms: 1.538618517 + 2.285714286 = 3.824332803 n = 3.824332803 Simplifying n = 3.824332803

Subproblem 2

n + -2.285714286 = -1.538618517 Simplifying n + -2.285714286 = -1.538618517 Reorder the terms: -2.285714286 + n = -1.538618517 Solving -2.285714286 + n = -1.538618517 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2.285714286' to each side of the equation. -2.285714286 + 2.285714286 + n = -1.538618517 + 2.285714286 Combine like terms: -2.285714286 + 2.285714286 = 0.000000000 0.000000000 + n = -1.538618517 + 2.285714286 n = -1.538618517 + 2.285714286 Combine like terms: -1.538618517 + 2.285714286 = 0.747095769 n = 0.747095769 Simplifying n = 0.747095769

Solution

The solution to the problem is based on the solutions from the subproblems. n = {3.824332803, 0.747095769}

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