105=1.75n^2-31.75n

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Solution for 105=1.75n^2-31.75n equation:


Simplifying
105 = 1.75n2 + -31.75n

Reorder the terms:
105 = -31.75n + 1.75n2

Solving
105 = -31.75n + 1.75n2

Solving for variable 'n'.

Reorder the terms:
105 + 31.75n + -1.75n2 = -31.75n + 31.75n + 1.75n2 + -1.75n2

Combine like terms: -31.75n + 31.75n = 0.00
105 + 31.75n + -1.75n2 = 0.00 + 1.75n2 + -1.75n2
105 + 31.75n + -1.75n2 = 1.75n2 + -1.75n2

Combine like terms: 1.75n2 + -1.75n2 = 0.00
105 + 31.75n + -1.75n2 = 0.00

Begin completing the square.  Divide all terms by
-1.75 the coefficient of the squared term: 

Divide each side by '-1.75'.
-60 + -18.14285714n + n2 = 0

Move the constant term to the right:

Add '60' to each side of the equation.
-60 + -18.14285714n + 60 + n2 = 0 + 60

Reorder the terms:
-60 + 60 + -18.14285714n + n2 = 0 + 60

Combine like terms: -60 + 60 = 0
0 + -18.14285714n + n2 = 0 + 60
-18.14285714n + n2 = 0 + 60

Combine like terms: 0 + 60 = 60
-18.14285714n + n2 = 60

The n term is -18.14285714n.  Take half its coefficient (-9.07142857).
Square it (82.29081630) and add it to both sides.

Add '82.29081630' to each side of the equation.
-18.14285714n + 82.29081630 + n2 = 60 + 82.29081630

Reorder the terms:
82.29081630 + -18.14285714n + n2 = 60 + 82.29081630

Combine like terms: 60 + 82.29081630 = 142.2908163
82.29081630 + -18.14285714n + n2 = 142.2908163

Factor a perfect square on the left side:
(n + -9.07142857)(n + -9.07142857) = 142.2908163

Calculate the square root of the right side: 11.928571427

Break this problem into two subproblems by setting 
(n + -9.07142857) equal to 11.928571427 and -11.928571427.

Subproblem 1

n + -9.07142857 = 11.928571427 Simplifying n + -9.07142857 = 11.928571427 Reorder the terms: -9.07142857 + n = 11.928571427 Solving -9.07142857 + n = 11.928571427 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '9.07142857' to each side of the equation. -9.07142857 + 9.07142857 + n = 11.928571427 + 9.07142857 Combine like terms: -9.07142857 + 9.07142857 = 0.00000000 0.00000000 + n = 11.928571427 + 9.07142857 n = 11.928571427 + 9.07142857 Combine like terms: 11.928571427 + 9.07142857 = 20.999999997 n = 20.999999997 Simplifying n = 20.999999997

Subproblem 2

n + -9.07142857 = -11.928571427 Simplifying n + -9.07142857 = -11.928571427 Reorder the terms: -9.07142857 + n = -11.928571427 Solving -9.07142857 + n = -11.928571427 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '9.07142857' to each side of the equation. -9.07142857 + 9.07142857 + n = -11.928571427 + 9.07142857 Combine like terms: -9.07142857 + 9.07142857 = 0.00000000 0.00000000 + n = -11.928571427 + 9.07142857 n = -11.928571427 + 9.07142857 Combine like terms: -11.928571427 + 9.07142857 = -2.857142857 n = -2.857142857 Simplifying n = -2.857142857

Solution

The solution to the problem is based on the solutions from the subproblems. n = {20.999999997, -2.857142857}

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