3x^2=5x-1

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


Simplifying
3x2 = 5x + -1

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

Solving
3x2 = -1 + 5x

Solving for variable 'x'.

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

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

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

Combine like terms: 5x + -5x = 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 = 0.3611111114
0.6944444447 + -1.666666667x + x2 = 0.3611111114

Factor a perfect square on the left side:
(x + -0.8333333335)(x + -0.8333333335) = 0.3611111114

Calculate the square root of the right side: 0.600925213

Break this problem into two subproblems by setting 
(x + -0.8333333335) equal to 0.600925213 and -0.600925213.

Subproblem 1

x + -0.8333333335 = 0.600925213 Simplifying x + -0.8333333335 = 0.600925213 Reorder the terms: -0.8333333335 + x = 0.600925213 Solving -0.8333333335 + x = 0.600925213 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 = 0.600925213 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + x = 0.600925213 + 0.8333333335 x = 0.600925213 + 0.8333333335 Combine like terms: 0.600925213 + 0.8333333335 = 1.4342585465 x = 1.4342585465 Simplifying x = 1.4342585465

Subproblem 2

x + -0.8333333335 = -0.600925213 Simplifying x + -0.8333333335 = -0.600925213 Reorder the terms: -0.8333333335 + x = -0.600925213 Solving -0.8333333335 + x = -0.600925213 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 = -0.600925213 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + x = -0.600925213 + 0.8333333335 x = -0.600925213 + 0.8333333335 Combine like terms: -0.600925213 + 0.8333333335 = 0.2324081205 x = 0.2324081205 Simplifying x = 0.2324081205

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.4342585465, 0.2324081205}

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