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

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


Simplifying
3x2 + -3x = x2 + 6

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

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

Solving
-3x + 3x2 = 6 + x2

Solving for variable 'x'.

Reorder the terms:
-6 + -3x + 3x2 + -1x2 = 6 + x2 + -6 + -1x2

Combine like terms: 3x2 + -1x2 = 2x2
-6 + -3x + 2x2 = 6 + x2 + -6 + -1x2

Reorder the terms:
-6 + -3x + 2x2 = 6 + -6 + x2 + -1x2

Combine like terms: 6 + -6 = 0
-6 + -3x + 2x2 = 0 + x2 + -1x2
-6 + -3x + 2x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
-6 + -3x + 2x2 = 0

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

Divide each side by '2'.
-3 + -1.5x + x2 = 0

Move the constant term to the right:

Add '3' to each side of the equation.
-3 + -1.5x + 3 + x2 = 0 + 3

Reorder the terms:
-3 + 3 + -1.5x + x2 = 0 + 3

Combine like terms: -3 + 3 = 0
0 + -1.5x + x2 = 0 + 3
-1.5x + x2 = 0 + 3

Combine like terms: 0 + 3 = 3
-1.5x + x2 = 3

The x term is -1.5x.  Take half its coefficient (-0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
-1.5x + 0.5625 + x2 = 3 + 0.5625

Reorder the terms:
0.5625 + -1.5x + x2 = 3 + 0.5625

Combine like terms: 3 + 0.5625 = 3.5625
0.5625 + -1.5x + x2 = 3.5625

Factor a perfect square on the left side:
(x + -0.75)(x + -0.75) = 3.5625

Calculate the square root of the right side: 1.887458609

Break this problem into two subproblems by setting 
(x + -0.75) equal to 1.887458609 and -1.887458609.

Subproblem 1

x + -0.75 = 1.887458609 Simplifying x + -0.75 = 1.887458609 Reorder the terms: -0.75 + x = 1.887458609 Solving -0.75 + x = 1.887458609 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + x = 1.887458609 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + x = 1.887458609 + 0.75 x = 1.887458609 + 0.75 Combine like terms: 1.887458609 + 0.75 = 2.637458609 x = 2.637458609 Simplifying x = 2.637458609

Subproblem 2

x + -0.75 = -1.887458609 Simplifying x + -0.75 = -1.887458609 Reorder the terms: -0.75 + x = -1.887458609 Solving -0.75 + x = -1.887458609 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + x = -1.887458609 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + x = -1.887458609 + 0.75 x = -1.887458609 + 0.75 Combine like terms: -1.887458609 + 0.75 = -1.137458609 x = -1.137458609 Simplifying x = -1.137458609

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.637458609, -1.137458609}

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