2x^2-6x=109

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Solution for 2x^2-6x=109 equation:


Simplifying
2x2 + -6x = 109

Reorder the terms:
-6x + 2x2 = 109

Solving
-6x + 2x2 = 109

Solving for variable 'x'.

Reorder the terms:
-109 + -6x + 2x2 = 109 + -109

Combine like terms: 109 + -109 = 0
-109 + -6x + 2x2 = 0

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

Divide each side by '2'.
-54.5 + -3x + x2 = 0

Move the constant term to the right:

Add '54.5' to each side of the equation.
-54.5 + -3x + 54.5 + x2 = 0 + 54.5

Reorder the terms:
-54.5 + 54.5 + -3x + x2 = 0 + 54.5

Combine like terms: -54.5 + 54.5 = 0.0
0.0 + -3x + x2 = 0 + 54.5
-3x + x2 = 0 + 54.5

Combine like terms: 0 + 54.5 = 54.5
-3x + x2 = 54.5

The x term is -3x.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3x + 2.25 + x2 = 54.5 + 2.25

Reorder the terms:
2.25 + -3x + x2 = 54.5 + 2.25

Combine like terms: 54.5 + 2.25 = 56.75
2.25 + -3x + x2 = 56.75

Factor a perfect square on the left side:
(x + -1.5)(x + -1.5) = 56.75

Calculate the square root of the right side: 7.533259587

Break this problem into two subproblems by setting 
(x + -1.5) equal to 7.533259587 and -7.533259587.

Subproblem 1

x + -1.5 = 7.533259587 Simplifying x + -1.5 = 7.533259587 Reorder the terms: -1.5 + x = 7.533259587 Solving -1.5 + x = 7.533259587 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + x = 7.533259587 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = 7.533259587 + 1.5 x = 7.533259587 + 1.5 Combine like terms: 7.533259587 + 1.5 = 9.033259587 x = 9.033259587 Simplifying x = 9.033259587

Subproblem 2

x + -1.5 = -7.533259587 Simplifying x + -1.5 = -7.533259587 Reorder the terms: -1.5 + x = -7.533259587 Solving -1.5 + x = -7.533259587 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + x = -7.533259587 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = -7.533259587 + 1.5 x = -7.533259587 + 1.5 Combine like terms: -7.533259587 + 1.5 = -6.033259587 x = -6.033259587 Simplifying x = -6.033259587

Solution

The solution to the problem is based on the solutions from the subproblems. x = {9.033259587, -6.033259587}

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