2n^2=10n-7

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Solution for 2n^2=10n-7 equation:


Simplifying
2n2 = 10n + -7

Reorder the terms:
2n2 = -7 + 10n

Solving
2n2 = -7 + 10n

Solving for variable 'n'.

Reorder the terms:
7 + -10n + 2n2 = -7 + 10n + 7 + -10n

Reorder the terms:
7 + -10n + 2n2 = -7 + 7 + 10n + -10n

Combine like terms: -7 + 7 = 0
7 + -10n + 2n2 = 0 + 10n + -10n
7 + -10n + 2n2 = 10n + -10n

Combine like terms: 10n + -10n = 0
7 + -10n + 2n2 = 0

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

Divide each side by '2'.
3.5 + -5n + n2 = 0

Move the constant term to the right:

Add '-3.5' to each side of the equation.
3.5 + -5n + -3.5 + n2 = 0 + -3.5

Reorder the terms:
3.5 + -3.5 + -5n + n2 = 0 + -3.5

Combine like terms: 3.5 + -3.5 = 0.0
0.0 + -5n + n2 = 0 + -3.5
-5n + n2 = 0 + -3.5

Combine like terms: 0 + -3.5 = -3.5
-5n + n2 = -3.5

The n term is -5n.  Take half its coefficient (-2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
-5n + 6.25 + n2 = -3.5 + 6.25

Reorder the terms:
6.25 + -5n + n2 = -3.5 + 6.25

Combine like terms: -3.5 + 6.25 = 2.75
6.25 + -5n + n2 = 2.75

Factor a perfect square on the left side:
(n + -2.5)(n + -2.5) = 2.75

Calculate the square root of the right side: 1.658312395

Break this problem into two subproblems by setting 
(n + -2.5) equal to 1.658312395 and -1.658312395.

Subproblem 1

n + -2.5 = 1.658312395 Simplifying n + -2.5 = 1.658312395 Reorder the terms: -2.5 + n = 1.658312395 Solving -2.5 + n = 1.658312395 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + n = 1.658312395 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + n = 1.658312395 + 2.5 n = 1.658312395 + 2.5 Combine like terms: 1.658312395 + 2.5 = 4.158312395 n = 4.158312395 Simplifying n = 4.158312395

Subproblem 2

n + -2.5 = -1.658312395 Simplifying n + -2.5 = -1.658312395 Reorder the terms: -2.5 + n = -1.658312395 Solving -2.5 + n = -1.658312395 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + n = -1.658312395 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + n = -1.658312395 + 2.5 n = -1.658312395 + 2.5 Combine like terms: -1.658312395 + 2.5 = 0.841687605 n = 0.841687605 Simplifying n = 0.841687605

Solution

The solution to the problem is based on the solutions from the subproblems. n = {4.158312395, 0.841687605}

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