5x=3x^2-1

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Solution for 5x=3x^2-1 equation:


Simplifying
5x = 3x2 + -1

Reorder the terms:
5x = -1 + 3x2

Solving
5x = -1 + 3x2

Solving for variable 'x'.

Reorder the terms:
1 + 5x + -3x2 = -1 + 3x2 + 1 + -3x2

Reorder the terms:
1 + 5x + -3x2 = -1 + 1 + 3x2 + -3x2

Combine like terms: -1 + 1 = 0
1 + 5x + -3x2 = 0 + 3x2 + -3x2
1 + 5x + -3x2 = 3x2 + -3x2

Combine like terms: 3x2 + -3x2 = 0
1 + 5x + -3x2 = 0

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

Divide each side by '-3'.
-0.3333333333 + -1.666666667x + x2 = 0

Move the constant term to the right:

Add '0.3333333333' to each side of the equation.
-0.3333333333 + -1.666666667x + 0.3333333333 + x2 = 0 + 0.3333333333

Reorder the terms:
-0.3333333333 + 0.3333333333 + -1.666666667x + x2 = 0 + 0.3333333333

Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000
0.0000000000 + -1.666666667x + x2 = 0 + 0.3333333333
-1.666666667x + x2 = 0 + 0.3333333333

Combine like terms: 0 + 0.3333333333 = 0.3333333333
-1.666666667x + x2 = 0.3333333333

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 = 0.3333333333 + 0.6944444447

Reorder the terms:
0.6944444447 + -1.666666667x + x2 = 0.3333333333 + 0.6944444447

Combine like terms: 0.3333333333 + 0.6944444447 = 1.027777778
0.6944444447 + -1.666666667x + x2 = 1.027777778

Factor a perfect square on the left side:
(x + -0.8333333335)(x + -0.8333333335) = 1.027777778

Calculate the square root of the right side: 1.013793755

Break this problem into two subproblems by setting 
(x + -0.8333333335) equal to 1.013793755 and -1.013793755.

Subproblem 1

x + -0.8333333335 = 1.013793755 Simplifying x + -0.8333333335 = 1.013793755 Reorder the terms: -0.8333333335 + x = 1.013793755 Solving -0.8333333335 + x = 1.013793755 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 = 1.013793755 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + x = 1.013793755 + 0.8333333335 x = 1.013793755 + 0.8333333335 Combine like terms: 1.013793755 + 0.8333333335 = 1.8471270885 x = 1.8471270885 Simplifying x = 1.8471270885

Subproblem 2

x + -0.8333333335 = -1.013793755 Simplifying x + -0.8333333335 = -1.013793755 Reorder the terms: -0.8333333335 + x = -1.013793755 Solving -0.8333333335 + x = -1.013793755 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 = -1.013793755 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + x = -1.013793755 + 0.8333333335 x = -1.013793755 + 0.8333333335 Combine like terms: -1.013793755 + 0.8333333335 = -0.1804604215 x = -0.1804604215 Simplifying x = -0.1804604215

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.8471270885, -0.1804604215}

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