6x^2-22x+9=0

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Solution for 6x^2-22x+9=0 equation:


Simplifying
6x2 + -22x + 9 = 0

Reorder the terms:
9 + -22x + 6x2 = 0

Solving
9 + -22x + 6x2 = 0

Solving for variable 'x'.

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

Divide each side by '6'.
1.5 + -3.666666667x + x2 = 0

Move the constant term to the right:

Add '-1.5' to each side of the equation.
1.5 + -3.666666667x + -1.5 + x2 = 0 + -1.5

Reorder the terms:
1.5 + -1.5 + -3.666666667x + x2 = 0 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + -3.666666667x + x2 = 0 + -1.5
-3.666666667x + x2 = 0 + -1.5

Combine like terms: 0 + -1.5 = -1.5
-3.666666667x + x2 = -1.5

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 = -1.5 + 3.361111114

Reorder the terms:
3.361111114 + -3.666666667x + x2 = -1.5 + 3.361111114

Combine like terms: -1.5 + 3.361111114 = 1.861111114
3.361111114 + -3.666666667x + x2 = 1.861111114

Factor a perfect square on the left side:
(x + -1.833333334)(x + -1.833333334) = 1.861111114

Calculate the square root of the right side: 1.364225463

Break this problem into two subproblems by setting 
(x + -1.833333334) equal to 1.364225463 and -1.364225463.

Subproblem 1

x + -1.833333334 = 1.364225463 Simplifying x + -1.833333334 = 1.364225463 Reorder the terms: -1.833333334 + x = 1.364225463 Solving -1.833333334 + x = 1.364225463 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 = 1.364225463 + 1.833333334 Combine like terms: -1.833333334 + 1.833333334 = 0.000000000 0.000000000 + x = 1.364225463 + 1.833333334 x = 1.364225463 + 1.833333334 Combine like terms: 1.364225463 + 1.833333334 = 3.197558797 x = 3.197558797 Simplifying x = 3.197558797

Subproblem 2

x + -1.833333334 = -1.364225463 Simplifying x + -1.833333334 = -1.364225463 Reorder the terms: -1.833333334 + x = -1.364225463 Solving -1.833333334 + x = -1.364225463 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 = -1.364225463 + 1.833333334 Combine like terms: -1.833333334 + 1.833333334 = 0.000000000 0.000000000 + x = -1.364225463 + 1.833333334 x = -1.364225463 + 1.833333334 Combine like terms: -1.364225463 + 1.833333334 = 0.469107871 x = 0.469107871 Simplifying x = 0.469107871

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.197558797, 0.469107871}

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