2n^2-10n=-9

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Solution for 2n^2-10n=-9 equation:


Simplifying
2n2 + -10n = -9

Reorder the terms:
-10n + 2n2 = -9

Solving
-10n + 2n2 = -9

Solving for variable 'n'.

Reorder the terms:
9 + -10n + 2n2 = -9 + 9

Combine like terms: -9 + 9 = 0
9 + -10n + 2n2 = 0

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

Divide each side by '2'.
4.5 + -5n + n2 = 0

Move the constant term to the right:

Add '-4.5' to each side of the equation.
4.5 + -5n + -4.5 + n2 = 0 + -4.5

Reorder the terms:
4.5 + -4.5 + -5n + n2 = 0 + -4.5

Combine like terms: 4.5 + -4.5 = 0.0
0.0 + -5n + n2 = 0 + -4.5
-5n + n2 = 0 + -4.5

Combine like terms: 0 + -4.5 = -4.5
-5n + n2 = -4.5

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 = -4.5 + 6.25

Reorder the terms:
6.25 + -5n + n2 = -4.5 + 6.25

Combine like terms: -4.5 + 6.25 = 1.75
6.25 + -5n + n2 = 1.75

Factor a perfect square on the left side:
(n + -2.5)(n + -2.5) = 1.75

Calculate the square root of the right side: 1.322875656

Break this problem into two subproblems by setting 
(n + -2.5) equal to 1.322875656 and -1.322875656.

Subproblem 1

n + -2.5 = 1.322875656 Simplifying n + -2.5 = 1.322875656 Reorder the terms: -2.5 + n = 1.322875656 Solving -2.5 + n = 1.322875656 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 = 1.322875656 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + n = 1.322875656 + 2.5 n = 1.322875656 + 2.5 Combine like terms: 1.322875656 + 2.5 = 3.822875656 n = 3.822875656 Simplifying n = 3.822875656

Subproblem 2

n + -2.5 = -1.322875656 Simplifying n + -2.5 = -1.322875656 Reorder the terms: -2.5 + n = -1.322875656 Solving -2.5 + n = -1.322875656 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 = -1.322875656 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + n = -1.322875656 + 2.5 n = -1.322875656 + 2.5 Combine like terms: -1.322875656 + 2.5 = 1.177124344 n = 1.177124344 Simplifying n = 1.177124344

Solution

The solution to the problem is based on the solutions from the subproblems. n = {3.822875656, 1.177124344}

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