3x=2x^2-1

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


Simplifying
3x = 2x2 + -1

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

Solving
3x = -1 + 2x2

Solving for variable 'x'.

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

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

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

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

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

Divide each side by '-2'.
-0.5 + -1.5x + x2 = 0

Move the constant term to the right:

Add '0.5' to each side of the equation.
-0.5 + -1.5x + 0.5 + x2 = 0 + 0.5

Reorder the terms:
-0.5 + 0.5 + -1.5x + x2 = 0 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + -1.5x + x2 = 0 + 0.5
-1.5x + x2 = 0 + 0.5

Combine like terms: 0 + 0.5 = 0.5
-1.5x + x2 = 0.5

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 = 0.5 + 0.5625

Reorder the terms:
0.5625 + -1.5x + x2 = 0.5 + 0.5625

Combine like terms: 0.5 + 0.5625 = 1.0625
0.5625 + -1.5x + x2 = 1.0625

Factor a perfect square on the left side:
(x + -0.75)(x + -0.75) = 1.0625

Calculate the square root of the right side: 1.030776406

Break this problem into two subproblems by setting 
(x + -0.75) equal to 1.030776406 and -1.030776406.

Subproblem 1

x + -0.75 = 1.030776406 Simplifying x + -0.75 = 1.030776406 Reorder the terms: -0.75 + x = 1.030776406 Solving -0.75 + x = 1.030776406 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 = 1.030776406 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + x = 1.030776406 + 0.75 x = 1.030776406 + 0.75 Combine like terms: 1.030776406 + 0.75 = 1.780776406 x = 1.780776406 Simplifying x = 1.780776406

Subproblem 2

x + -0.75 = -1.030776406 Simplifying x + -0.75 = -1.030776406 Reorder the terms: -0.75 + x = -1.030776406 Solving -0.75 + x = -1.030776406 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 = -1.030776406 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + x = -1.030776406 + 0.75 x = -1.030776406 + 0.75 Combine like terms: -1.030776406 + 0.75 = -0.280776406 x = -0.280776406 Simplifying x = -0.280776406

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.780776406, -0.280776406}

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