2n^2=7n-1

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Solution for 2n^2=7n-1 equation:


Simplifying
2n2 = 7n + -1

Reorder the terms:
2n2 = -1 + 7n

Solving
2n2 = -1 + 7n

Solving for variable 'n'.

Reorder the terms:
1 + -7n + 2n2 = -1 + 7n + 1 + -7n

Reorder the terms:
1 + -7n + 2n2 = -1 + 1 + 7n + -7n

Combine like terms: -1 + 1 = 0
1 + -7n + 2n2 = 0 + 7n + -7n
1 + -7n + 2n2 = 7n + -7n

Combine like terms: 7n + -7n = 0
1 + -7n + 2n2 = 0

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

Divide each side by '2'.
0.5 + -3.5n + n2 = 0

Move the constant term to the right:

Add '-0.5' to each side of the equation.
0.5 + -3.5n + -0.5 + n2 = 0 + -0.5

Reorder the terms:
0.5 + -0.5 + -3.5n + n2 = 0 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + -3.5n + n2 = 0 + -0.5
-3.5n + n2 = 0 + -0.5

Combine like terms: 0 + -0.5 = -0.5
-3.5n + n2 = -0.5

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 = -0.5 + 3.0625

Reorder the terms:
3.0625 + -3.5n + n2 = -0.5 + 3.0625

Combine like terms: -0.5 + 3.0625 = 2.5625
3.0625 + -3.5n + n2 = 2.5625

Factor a perfect square on the left side:
(n + -1.75)(n + -1.75) = 2.5625

Calculate the square root of the right side: 1.600781059

Break this problem into two subproblems by setting 
(n + -1.75) equal to 1.600781059 and -1.600781059.

Subproblem 1

n + -1.75 = 1.600781059 Simplifying n + -1.75 = 1.600781059 Reorder the terms: -1.75 + n = 1.600781059 Solving -1.75 + n = 1.600781059 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 = 1.600781059 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + n = 1.600781059 + 1.75 n = 1.600781059 + 1.75 Combine like terms: 1.600781059 + 1.75 = 3.350781059 n = 3.350781059 Simplifying n = 3.350781059

Subproblem 2

n + -1.75 = -1.600781059 Simplifying n + -1.75 = -1.600781059 Reorder the terms: -1.75 + n = -1.600781059 Solving -1.75 + n = -1.600781059 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 = -1.600781059 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + n = -1.600781059 + 1.75 n = -1.600781059 + 1.75 Combine like terms: -1.600781059 + 1.75 = 0.149218941 n = 0.149218941 Simplifying n = 0.149218941

Solution

The solution to the problem is based on the solutions from the subproblems. n = {3.350781059, 0.149218941}

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