3x(x-1)=2x+9

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Solution for 3x(x-1)=2x+9 equation:


Simplifying
3x(x + -1) = 2x + 9

Reorder the terms:
3x(-1 + x) = 2x + 9
(-1 * 3x + x * 3x) = 2x + 9
(-3x + 3x2) = 2x + 9

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

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

Solving for variable 'x'.

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

Combine like terms: -3x + -2x = -5x
-9 + -5x + 3x2 = 9 + 2x + -9 + -2x

Reorder the terms:
-9 + -5x + 3x2 = 9 + -9 + 2x + -2x

Combine like terms: 9 + -9 = 0
-9 + -5x + 3x2 = 0 + 2x + -2x
-9 + -5x + 3x2 = 2x + -2x

Combine like terms: 2x + -2x = 0
-9 + -5x + 3x2 = 0

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

Divide each side by '3'.
-3 + -1.666666667x + x2 = 0

Move the constant term to the right:

Add '3' to each side of the equation.
-3 + -1.666666667x + 3 + x2 = 0 + 3

Reorder the terms:
-3 + 3 + -1.666666667x + x2 = 0 + 3

Combine like terms: -3 + 3 = 0
0 + -1.666666667x + x2 = 0 + 3
-1.666666667x + x2 = 0 + 3

Combine like terms: 0 + 3 = 3
-1.666666667x + x2 = 3

The x term is -1.666666667x.  Take half its coefficient (-0.8333333335).
Square it (0.6944444447) and add it to both sides.

Add '0.6944444447' to each side of the equation.
-1.666666667x + 0.6944444447 + x2 = 3 + 0.6944444447

Reorder the terms:
0.6944444447 + -1.666666667x + x2 = 3 + 0.6944444447

Combine like terms: 3 + 0.6944444447 = 3.6944444447
0.6944444447 + -1.666666667x + x2 = 3.6944444447

Factor a perfect square on the left side:
(x + -0.8333333335)(x + -0.8333333335) = 3.6944444447

Calculate the square root of the right side: 1.922093766

Break this problem into two subproblems by setting 
(x + -0.8333333335) equal to 1.922093766 and -1.922093766.

Subproblem 1

x + -0.8333333335 = 1.922093766 Simplifying x + -0.8333333335 = 1.922093766 Reorder the terms: -0.8333333335 + x = 1.922093766 Solving -0.8333333335 + x = 1.922093766 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + x = 1.922093766 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + x = 1.922093766 + 0.8333333335 x = 1.922093766 + 0.8333333335 Combine like terms: 1.922093766 + 0.8333333335 = 2.7554270995 x = 2.7554270995 Simplifying x = 2.7554270995

Subproblem 2

x + -0.8333333335 = -1.922093766 Simplifying x + -0.8333333335 = -1.922093766 Reorder the terms: -0.8333333335 + x = -1.922093766 Solving -0.8333333335 + x = -1.922093766 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + x = -1.922093766 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + x = -1.922093766 + 0.8333333335 x = -1.922093766 + 0.8333333335 Combine like terms: -1.922093766 + 0.8333333335 = -1.0887604325 x = -1.0887604325 Simplifying x = -1.0887604325

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.7554270995, -1.0887604325}

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