3x^2-2x+6=9x+2

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Solution for 3x^2-2x+6=9x+2 equation:


Simplifying
3x2 + -2x + 6 = 9x + 2

Reorder the terms:
6 + -2x + 3x2 = 9x + 2

Reorder the terms:
6 + -2x + 3x2 = 2 + 9x

Solving
6 + -2x + 3x2 = 2 + 9x

Solving for variable 'x'.

Reorder the terms:
6 + -2 + -2x + -9x + 3x2 = 2 + 9x + -2 + -9x

Combine like terms: 6 + -2 = 4
4 + -2x + -9x + 3x2 = 2 + 9x + -2 + -9x

Combine like terms: -2x + -9x = -11x
4 + -11x + 3x2 = 2 + 9x + -2 + -9x

Reorder the terms:
4 + -11x + 3x2 = 2 + -2 + 9x + -9x

Combine like terms: 2 + -2 = 0
4 + -11x + 3x2 = 0 + 9x + -9x
4 + -11x + 3x2 = 9x + -9x

Combine like terms: 9x + -9x = 0
4 + -11x + 3x2 = 0

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

Divide each side by '3'.
1.333333333 + -3.666666667x + x2 = 0

Move the constant term to the right:

Add '-1.333333333' to each side of the equation.
1.333333333 + -3.666666667x + -1.333333333 + x2 = 0 + -1.333333333

Reorder the terms:
1.333333333 + -1.333333333 + -3.666666667x + x2 = 0 + -1.333333333

Combine like terms: 1.333333333 + -1.333333333 = 0.000000000
0.000000000 + -3.666666667x + x2 = 0 + -1.333333333
-3.666666667x + x2 = 0 + -1.333333333

Combine like terms: 0 + -1.333333333 = -1.333333333
-3.666666667x + x2 = -1.333333333

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 = -1.333333333 + 3.361111114

Reorder the terms:
3.361111114 + -3.666666667x + x2 = -1.333333333 + 3.361111114

Combine like terms: -1.333333333 + 3.361111114 = 2.027777781
3.361111114 + -3.666666667x + x2 = 2.027777781

Factor a perfect square on the left side:
(x + -1.833333334)(x + -1.833333334) = 2.027777781

Calculate the square root of the right side: 1.424000625

Break this problem into two subproblems by setting 
(x + -1.833333334) equal to 1.424000625 and -1.424000625.

Subproblem 1

x + -1.833333334 = 1.424000625 Simplifying x + -1.833333334 = 1.424000625 Reorder the terms: -1.833333334 + x = 1.424000625 Solving -1.833333334 + x = 1.424000625 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 = 1.424000625 + 1.833333334 Combine like terms: -1.833333334 + 1.833333334 = 0.000000000 0.000000000 + x = 1.424000625 + 1.833333334 x = 1.424000625 + 1.833333334 Combine like terms: 1.424000625 + 1.833333334 = 3.257333959 x = 3.257333959 Simplifying x = 3.257333959

Subproblem 2

x + -1.833333334 = -1.424000625 Simplifying x + -1.833333334 = -1.424000625 Reorder the terms: -1.833333334 + x = -1.424000625 Solving -1.833333334 + x = -1.424000625 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 = -1.424000625 + 1.833333334 Combine like terms: -1.833333334 + 1.833333334 = 0.000000000 0.000000000 + x = -1.424000625 + 1.833333334 x = -1.424000625 + 1.833333334 Combine like terms: -1.424000625 + 1.833333334 = 0.409332709 x = 0.409332709 Simplifying x = 0.409332709

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.257333959, 0.409332709}

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