3x^2-11x+6=65

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Solution for 3x^2-11x+6=65 equation:


Simplifying
3x2 + -11x + 6 = 65

Reorder the terms:
6 + -11x + 3x2 = 65

Solving
6 + -11x + 3x2 = 65

Solving for variable 'x'.

Reorder the terms:
6 + -65 + -11x + 3x2 = 65 + -65

Combine like terms: 6 + -65 = -59
-59 + -11x + 3x2 = 65 + -65

Combine like terms: 65 + -65 = 0
-59 + -11x + 3x2 = 0

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

Divide each side by '3'.
-19.66666667 + -3.666666667x + x2 = 0

Move the constant term to the right:

Add '19.66666667' to each side of the equation.
-19.66666667 + -3.666666667x + 19.66666667 + x2 = 0 + 19.66666667

Reorder the terms:
-19.66666667 + 19.66666667 + -3.666666667x + x2 = 0 + 19.66666667

Combine like terms: -19.66666667 + 19.66666667 = 0.00000000
0.00000000 + -3.666666667x + x2 = 0 + 19.66666667
-3.666666667x + x2 = 0 + 19.66666667

Combine like terms: 0 + 19.66666667 = 19.66666667
-3.666666667x + x2 = 19.66666667

The x term is -3.666666667x.  Take half its coefficient (-1.833333334).
Square it (3.361111114) and add it to both sides.

Add '3.361111114' to each side of the equation.
-3.666666667x + 3.361111114 + x2 = 19.66666667 + 3.361111114

Reorder the terms:
3.361111114 + -3.666666667x + x2 = 19.66666667 + 3.361111114

Combine like terms: 19.66666667 + 3.361111114 = 23.027777784
3.361111114 + -3.666666667x + x2 = 23.027777784

Factor a perfect square on the left side:
(x + -1.833333334)(x + -1.833333334) = 23.027777784

Calculate the square root of the right side: 4.798726684

Break this problem into two subproblems by setting 
(x + -1.833333334) equal to 4.798726684 and -4.798726684.

Subproblem 1

x + -1.833333334 = 4.798726684 Simplifying x + -1.833333334 = 4.798726684 Reorder the terms: -1.833333334 + x = 4.798726684 Solving -1.833333334 + x = 4.798726684 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.833333334' to each side of the equation. -1.833333334 + 1.833333334 + x = 4.798726684 + 1.833333334 Combine like terms: -1.833333334 + 1.833333334 = 0.000000000 0.000000000 + x = 4.798726684 + 1.833333334 x = 4.798726684 + 1.833333334 Combine like terms: 4.798726684 + 1.833333334 = 6.632060018 x = 6.632060018 Simplifying x = 6.632060018

Subproblem 2

x + -1.833333334 = -4.798726684 Simplifying x + -1.833333334 = -4.798726684 Reorder the terms: -1.833333334 + x = -4.798726684 Solving -1.833333334 + x = -4.798726684 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.833333334' to each side of the equation. -1.833333334 + 1.833333334 + x = -4.798726684 + 1.833333334 Combine like terms: -1.833333334 + 1.833333334 = 0.000000000 0.000000000 + x = -4.798726684 + 1.833333334 x = -4.798726684 + 1.833333334 Combine like terms: -4.798726684 + 1.833333334 = -2.96539335 x = -2.96539335 Simplifying x = -2.96539335

Solution

The solution to the problem is based on the solutions from the subproblems. x = {6.632060018, -2.96539335}

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