3x^2-83x=105

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Solution for 3x^2-83x=105 equation:


Simplifying
3x2 + -83x = 105

Reorder the terms:
-83x + 3x2 = 105

Solving
-83x + 3x2 = 105

Solving for variable 'x'.

Reorder the terms:
-105 + -83x + 3x2 = 105 + -105

Combine like terms: 105 + -105 = 0
-105 + -83x + 3x2 = 0

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

Divide each side by '3'.
-35 + -27.66666667x + x2 = 0

Move the constant term to the right:

Add '35' to each side of the equation.
-35 + -27.66666667x + 35 + x2 = 0 + 35

Reorder the terms:
-35 + 35 + -27.66666667x + x2 = 0 + 35

Combine like terms: -35 + 35 = 0
0 + -27.66666667x + x2 = 0 + 35
-27.66666667x + x2 = 0 + 35

Combine like terms: 0 + 35 = 35
-27.66666667x + x2 = 35

The x term is -27.66666667x.  Take half its coefficient (-13.83333334).
Square it (191.3611113) and add it to both sides.

Add '191.3611113' to each side of the equation.
-27.66666667x + 191.3611113 + x2 = 35 + 191.3611113

Reorder the terms:
191.3611113 + -27.66666667x + x2 = 35 + 191.3611113

Combine like terms: 35 + 191.3611113 = 226.3611113
191.3611113 + -27.66666667x + x2 = 226.3611113

Factor a perfect square on the left side:
(x + -13.83333334)(x + -13.83333334) = 226.3611113

Calculate the square root of the right side: 15.045301968

Break this problem into two subproblems by setting 
(x + -13.83333334) equal to 15.045301968 and -15.045301968.

Subproblem 1

x + -13.83333334 = 15.045301968 Simplifying x + -13.83333334 = 15.045301968 Reorder the terms: -13.83333334 + x = 15.045301968 Solving -13.83333334 + x = 15.045301968 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '13.83333334' to each side of the equation. -13.83333334 + 13.83333334 + x = 15.045301968 + 13.83333334 Combine like terms: -13.83333334 + 13.83333334 = 0.00000000 0.00000000 + x = 15.045301968 + 13.83333334 x = 15.045301968 + 13.83333334 Combine like terms: 15.045301968 + 13.83333334 = 28.878635308 x = 28.878635308 Simplifying x = 28.878635308

Subproblem 2

x + -13.83333334 = -15.045301968 Simplifying x + -13.83333334 = -15.045301968 Reorder the terms: -13.83333334 + x = -15.045301968 Solving -13.83333334 + x = -15.045301968 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '13.83333334' to each side of the equation. -13.83333334 + 13.83333334 + x = -15.045301968 + 13.83333334 Combine like terms: -13.83333334 + 13.83333334 = 0.00000000 0.00000000 + x = -15.045301968 + 13.83333334 x = -15.045301968 + 13.83333334 Combine like terms: -15.045301968 + 13.83333334 = -1.211968628 x = -1.211968628 Simplifying x = -1.211968628

Solution

The solution to the problem is based on the solutions from the subproblems. x = {28.878635308, -1.211968628}

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