2-x^2=3x(x-1)

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


Simplifying
2 + -1x2 = 3x(x + -1)

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

Solving
2 + -1x2 = -3x + 3x2

Solving for variable 'x'.

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

Combine like terms: -1x2 + -3x2 = -4x2
2 + 3x + -4x2 = -3x + 3x2 + 3x + -3x2

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

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

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

Begin completing the square.  Divide all terms by
-4 the coefficient of the squared term: 

Divide each side by '-4'.
-0.5 + -0.75x + x2 = 0

Move the constant term to the right:

Add '0.5' to each side of the equation.
-0.5 + -0.75x + 0.5 + x2 = 0 + 0.5

Reorder the terms:
-0.5 + 0.5 + -0.75x + x2 = 0 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + -0.75x + x2 = 0 + 0.5
-0.75x + x2 = 0 + 0.5

Combine like terms: 0 + 0.5 = 0.5
-0.75x + x2 = 0.5

The x term is -0.75x.  Take half its coefficient (-0.375).
Square it (0.140625) and add it to both sides.

Add '0.140625' to each side of the equation.
-0.75x + 0.140625 + x2 = 0.5 + 0.140625

Reorder the terms:
0.140625 + -0.75x + x2 = 0.5 + 0.140625

Combine like terms: 0.5 + 0.140625 = 0.640625
0.140625 + -0.75x + x2 = 0.640625

Factor a perfect square on the left side:
(x + -0.375)(x + -0.375) = 0.640625

Calculate the square root of the right side: 0.80039053

Break this problem into two subproblems by setting 
(x + -0.375) equal to 0.80039053 and -0.80039053.

Subproblem 1

x + -0.375 = 0.80039053 Simplifying x + -0.375 = 0.80039053 Reorder the terms: -0.375 + x = 0.80039053 Solving -0.375 + x = 0.80039053 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + x = 0.80039053 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + x = 0.80039053 + 0.375 x = 0.80039053 + 0.375 Combine like terms: 0.80039053 + 0.375 = 1.17539053 x = 1.17539053 Simplifying x = 1.17539053

Subproblem 2

x + -0.375 = -0.80039053 Simplifying x + -0.375 = -0.80039053 Reorder the terms: -0.375 + x = -0.80039053 Solving -0.375 + x = -0.80039053 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + x = -0.80039053 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + x = -0.80039053 + 0.375 x = -0.80039053 + 0.375 Combine like terms: -0.80039053 + 0.375 = -0.42539053 x = -0.42539053 Simplifying x = -0.42539053

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.17539053, -0.42539053}

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