0=3x^2-20x+21

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Solution for 0=3x^2-20x+21 equation:


Simplifying
0 = 3x2 + -20x + 21

Reorder the terms:
0 = 21 + -20x + 3x2

Solving
0 = 21 + -20x + 3x2

Solving for variable 'x'.

Combine like terms: 0 + -21 = -21
-21 + 20x + -3x2 = 21 + -20x + 3x2 + -21 + 20x + -3x2

Reorder the terms:
-21 + 20x + -3x2 = 21 + -21 + -20x + 20x + 3x2 + -3x2

Combine like terms: 21 + -21 = 0
-21 + 20x + -3x2 = 0 + -20x + 20x + 3x2 + -3x2
-21 + 20x + -3x2 = -20x + 20x + 3x2 + -3x2

Combine like terms: -20x + 20x = 0
-21 + 20x + -3x2 = 0 + 3x2 + -3x2
-21 + 20x + -3x2 = 3x2 + -3x2

Combine like terms: 3x2 + -3x2 = 0
-21 + 20x + -3x2 = 0

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

Divide each side by '-3'.
7 + -6.666666667x + x2 = 0

Move the constant term to the right:

Add '-7' to each side of the equation.
7 + -6.666666667x + -7 + x2 = 0 + -7

Reorder the terms:
7 + -7 + -6.666666667x + x2 = 0 + -7

Combine like terms: 7 + -7 = 0
0 + -6.666666667x + x2 = 0 + -7
-6.666666667x + x2 = 0 + -7

Combine like terms: 0 + -7 = -7
-6.666666667x + x2 = -7

The x term is -6.666666667x.  Take half its coefficient (-3.333333334).
Square it (11.11111112) and add it to both sides.

Add '11.11111112' to each side of the equation.
-6.666666667x + 11.11111112 + x2 = -7 + 11.11111112

Reorder the terms:
11.11111112 + -6.666666667x + x2 = -7 + 11.11111112

Combine like terms: -7 + 11.11111112 = 4.11111112
11.11111112 + -6.666666667x + x2 = 4.11111112

Factor a perfect square on the left side:
(x + -3.333333334)(x + -3.333333334) = 4.11111112

Calculate the square root of the right side: 2.027587512

Break this problem into two subproblems by setting 
(x + -3.333333334) equal to 2.027587512 and -2.027587512.

Subproblem 1

x + -3.333333334 = 2.027587512 Simplifying x + -3.333333334 = 2.027587512 Reorder the terms: -3.333333334 + x = 2.027587512 Solving -3.333333334 + x = 2.027587512 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.333333334' to each side of the equation. -3.333333334 + 3.333333334 + x = 2.027587512 + 3.333333334 Combine like terms: -3.333333334 + 3.333333334 = 0.000000000 0.000000000 + x = 2.027587512 + 3.333333334 x = 2.027587512 + 3.333333334 Combine like terms: 2.027587512 + 3.333333334 = 5.360920846 x = 5.360920846 Simplifying x = 5.360920846

Subproblem 2

x + -3.333333334 = -2.027587512 Simplifying x + -3.333333334 = -2.027587512 Reorder the terms: -3.333333334 + x = -2.027587512 Solving -3.333333334 + x = -2.027587512 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.333333334' to each side of the equation. -3.333333334 + 3.333333334 + x = -2.027587512 + 3.333333334 Combine like terms: -3.333333334 + 3.333333334 = 0.000000000 0.000000000 + x = -2.027587512 + 3.333333334 x = -2.027587512 + 3.333333334 Combine like terms: -2.027587512 + 3.333333334 = 1.305745822 x = 1.305745822 Simplifying x = 1.305745822

Solution

The solution to the problem is based on the solutions from the subproblems. x = {5.360920846, 1.305745822}

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