3x^2+5x-3=x

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


Simplifying
3x2 + 5x + -3 = x

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

Solving
-3 + 5x + 3x2 = x

Solving for variable 'x'.

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

Combine like terms: 5x + -1x = 4x
-3 + 4x + 3x2 = x + -1x

Combine like terms: x + -1x = 0
-3 + 4x + 3x2 = 0

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

Divide each side by '3'.
-1 + 1.333333333x + x2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + 1.333333333x + 1 + x2 = 0 + 1

Reorder the terms:
-1 + 1 + 1.333333333x + x2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + 1.333333333x + x2 = 0 + 1
1.333333333x + x2 = 0 + 1

Combine like terms: 0 + 1 = 1
1.333333333x + x2 = 1

The x term is 1.333333333x.  Take half its coefficient (0.6666666665).
Square it (0.4444444442) and add it to both sides.

Add '0.4444444442' to each side of the equation.
1.333333333x + 0.4444444442 + x2 = 1 + 0.4444444442

Reorder the terms:
0.4444444442 + 1.333333333x + x2 = 1 + 0.4444444442

Combine like terms: 1 + 0.4444444442 = 1.4444444442
0.4444444442 + 1.333333333x + x2 = 1.4444444442

Factor a perfect square on the left side:
(x + 0.6666666665)(x + 0.6666666665) = 1.4444444442

Calculate the square root of the right side: 1.201850425

Break this problem into two subproblems by setting 
(x + 0.6666666665) equal to 1.201850425 and -1.201850425.

Subproblem 1

x + 0.6666666665 = 1.201850425 Simplifying x + 0.6666666665 = 1.201850425 Reorder the terms: 0.6666666665 + x = 1.201850425 Solving 0.6666666665 + x = 1.201850425 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.6666666665' to each side of the equation. 0.6666666665 + -0.6666666665 + x = 1.201850425 + -0.6666666665 Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000 0.0000000000 + x = 1.201850425 + -0.6666666665 x = 1.201850425 + -0.6666666665 Combine like terms: 1.201850425 + -0.6666666665 = 0.5351837585 x = 0.5351837585 Simplifying x = 0.5351837585

Subproblem 2

x + 0.6666666665 = -1.201850425 Simplifying x + 0.6666666665 = -1.201850425 Reorder the terms: 0.6666666665 + x = -1.201850425 Solving 0.6666666665 + x = -1.201850425 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.6666666665' to each side of the equation. 0.6666666665 + -0.6666666665 + x = -1.201850425 + -0.6666666665 Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000 0.0000000000 + x = -1.201850425 + -0.6666666665 x = -1.201850425 + -0.6666666665 Combine like terms: -1.201850425 + -0.6666666665 = -1.8685170915 x = -1.8685170915 Simplifying x = -1.8685170915

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.5351837585, -1.8685170915}

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