8x=3x^2-1

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Solution for 8x=3x^2-1 equation:


Simplifying
8x = 3x2 + -1

Reorder the terms:
8x = -1 + 3x2

Solving
8x = -1 + 3x2

Solving for variable 'x'.

Reorder the terms:
1 + 8x + -3x2 = -1 + 3x2 + 1 + -3x2

Reorder the terms:
1 + 8x + -3x2 = -1 + 1 + 3x2 + -3x2

Combine like terms: -1 + 1 = 0
1 + 8x + -3x2 = 0 + 3x2 + -3x2
1 + 8x + -3x2 = 3x2 + -3x2

Combine like terms: 3x2 + -3x2 = 0
1 + 8x + -3x2 = 0

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

Divide each side by '-3'.
-0.3333333333 + -2.666666667x + x2 = 0

Move the constant term to the right:

Add '0.3333333333' to each side of the equation.
-0.3333333333 + -2.666666667x + 0.3333333333 + x2 = 0 + 0.3333333333

Reorder the terms:
-0.3333333333 + 0.3333333333 + -2.666666667x + x2 = 0 + 0.3333333333

Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000
0.0000000000 + -2.666666667x + x2 = 0 + 0.3333333333
-2.666666667x + x2 = 0 + 0.3333333333

Combine like terms: 0 + 0.3333333333 = 0.3333333333
-2.666666667x + x2 = 0.3333333333

The x term is -2.666666667x.  Take half its coefficient (-1.333333334).
Square it (1.777777780) and add it to both sides.

Add '1.777777780' to each side of the equation.
-2.666666667x + 1.777777780 + x2 = 0.3333333333 + 1.777777780

Reorder the terms:
1.777777780 + -2.666666667x + x2 = 0.3333333333 + 1.777777780

Combine like terms: 0.3333333333 + 1.777777780 = 2.1111111133
1.777777780 + -2.666666667x + x2 = 2.1111111133

Factor a perfect square on the left side:
(x + -1.333333334)(x + -1.333333334) = 2.1111111133

Calculate the square root of the right side: 1.452966315

Break this problem into two subproblems by setting 
(x + -1.333333334) equal to 1.452966315 and -1.452966315.

Subproblem 1

x + -1.333333334 = 1.452966315 Simplifying x + -1.333333334 = 1.452966315 Reorder the terms: -1.333333334 + x = 1.452966315 Solving -1.333333334 + x = 1.452966315 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.333333334' to each side of the equation. -1.333333334 + 1.333333334 + x = 1.452966315 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + x = 1.452966315 + 1.333333334 x = 1.452966315 + 1.333333334 Combine like terms: 1.452966315 + 1.333333334 = 2.786299649 x = 2.786299649 Simplifying x = 2.786299649

Subproblem 2

x + -1.333333334 = -1.452966315 Simplifying x + -1.333333334 = -1.452966315 Reorder the terms: -1.333333334 + x = -1.452966315 Solving -1.333333334 + x = -1.452966315 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.333333334' to each side of the equation. -1.333333334 + 1.333333334 + x = -1.452966315 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + x = -1.452966315 + 1.333333334 x = -1.452966315 + 1.333333334 Combine like terms: -1.452966315 + 1.333333334 = -0.119632981 x = -0.119632981 Simplifying x = -0.119632981

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.786299649, -0.119632981}

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