0=3x^2-7x+3

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Solution for 0=3x^2-7x+3 equation:


Simplifying
0 = 3x2 + -7x + 3

Reorder the terms:
0 = 3 + -7x + 3x2

Solving
0 = 3 + -7x + 3x2

Solving for variable 'x'.

Combine like terms: 0 + -3 = -3
-3 + 7x + -3x2 = 3 + -7x + 3x2 + -3 + 7x + -3x2

Reorder the terms:
-3 + 7x + -3x2 = 3 + -3 + -7x + 7x + 3x2 + -3x2

Combine like terms: 3 + -3 = 0
-3 + 7x + -3x2 = 0 + -7x + 7x + 3x2 + -3x2
-3 + 7x + -3x2 = -7x + 7x + 3x2 + -3x2

Combine like terms: -7x + 7x = 0
-3 + 7x + -3x2 = 0 + 3x2 + -3x2
-3 + 7x + -3x2 = 3x2 + -3x2

Combine like terms: 3x2 + -3x2 = 0
-3 + 7x + -3x2 = 0

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

Divide each side by '-3'.
1 + -2.333333333x + x2 = 0

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + -2.333333333x + -1 + x2 = 0 + -1

Reorder the terms:
1 + -1 + -2.333333333x + x2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + -2.333333333x + x2 = 0 + -1
-2.333333333x + x2 = 0 + -1

Combine like terms: 0 + -1 = -1
-2.333333333x + x2 = -1

The x term is -2.333333333x.  Take half its coefficient (-1.166666667).
Square it (1.361111112) and add it to both sides.

Add '1.361111112' to each side of the equation.
-2.333333333x + 1.361111112 + x2 = -1 + 1.361111112

Reorder the terms:
1.361111112 + -2.333333333x + x2 = -1 + 1.361111112

Combine like terms: -1 + 1.361111112 = 0.361111112
1.361111112 + -2.333333333x + x2 = 0.361111112

Factor a perfect square on the left side:
(x + -1.166666667)(x + -1.166666667) = 0.361111112

Calculate the square root of the right side: 0.600925213

Break this problem into two subproblems by setting 
(x + -1.166666667) equal to 0.600925213 and -0.600925213.

Subproblem 1

x + -1.166666667 = 0.600925213 Simplifying x + -1.166666667 = 0.600925213 Reorder the terms: -1.166666667 + x = 0.600925213 Solving -1.166666667 + x = 0.600925213 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.166666667' to each side of the equation. -1.166666667 + 1.166666667 + x = 0.600925213 + 1.166666667 Combine like terms: -1.166666667 + 1.166666667 = 0.000000000 0.000000000 + x = 0.600925213 + 1.166666667 x = 0.600925213 + 1.166666667 Combine like terms: 0.600925213 + 1.166666667 = 1.76759188 x = 1.76759188 Simplifying x = 1.76759188

Subproblem 2

x + -1.166666667 = -0.600925213 Simplifying x + -1.166666667 = -0.600925213 Reorder the terms: -1.166666667 + x = -0.600925213 Solving -1.166666667 + x = -0.600925213 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.166666667' to each side of the equation. -1.166666667 + 1.166666667 + x = -0.600925213 + 1.166666667 Combine like terms: -1.166666667 + 1.166666667 = 0.000000000 0.000000000 + x = -0.600925213 + 1.166666667 x = -0.600925213 + 1.166666667 Combine like terms: -0.600925213 + 1.166666667 = 0.565741454 x = 0.565741454 Simplifying x = 0.565741454

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.76759188, 0.565741454}

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