3x(21-x)=x(3x-2)-24

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Solution for 3x(21-x)=x(3x-2)-24 equation:


Simplifying
3x(21 + -1x) = x(3x + -2) + -24
(21 * 3x + -1x * 3x) = x(3x + -2) + -24
(63x + -3x2) = x(3x + -2) + -24

Reorder the terms:
63x + -3x2 = x(-2 + 3x) + -24
63x + -3x2 = (-2 * x + 3x * x) + -24
63x + -3x2 = (-2x + 3x2) + -24

Reorder the terms:
63x + -3x2 = -24 + -2x + 3x2

Solving
63x + -3x2 = -24 + -2x + 3x2

Solving for variable 'x'.

Reorder the terms:
24 + 63x + 2x + -3x2 + -3x2 = -24 + -2x + 3x2 + 24 + 2x + -3x2

Combine like terms: 63x + 2x = 65x
24 + 65x + -3x2 + -3x2 = -24 + -2x + 3x2 + 24 + 2x + -3x2

Combine like terms: -3x2 + -3x2 = -6x2
24 + 65x + -6x2 = -24 + -2x + 3x2 + 24 + 2x + -3x2

Reorder the terms:
24 + 65x + -6x2 = -24 + 24 + -2x + 2x + 3x2 + -3x2

Combine like terms: -24 + 24 = 0
24 + 65x + -6x2 = 0 + -2x + 2x + 3x2 + -3x2
24 + 65x + -6x2 = -2x + 2x + 3x2 + -3x2

Combine like terms: -2x + 2x = 0
24 + 65x + -6x2 = 0 + 3x2 + -3x2
24 + 65x + -6x2 = 3x2 + -3x2

Combine like terms: 3x2 + -3x2 = 0
24 + 65x + -6x2 = 0

Begin completing the square.  Divide all terms by
-6 the coefficient of the squared term: 

Divide each side by '-6'.
-4 + -10.83333333x + x2 = 0

Move the constant term to the right:

Add '4' to each side of the equation.
-4 + -10.83333333x + 4 + x2 = 0 + 4

Reorder the terms:
-4 + 4 + -10.83333333x + x2 = 0 + 4

Combine like terms: -4 + 4 = 0
0 + -10.83333333x + x2 = 0 + 4
-10.83333333x + x2 = 0 + 4

Combine like terms: 0 + 4 = 4
-10.83333333x + x2 = 4

The x term is -10.83333333x.  Take half its coefficient (-5.416666665).
Square it (29.34027776) and add it to both sides.

Add '29.34027776' to each side of the equation.
-10.83333333x + 29.34027776 + x2 = 4 + 29.34027776

Reorder the terms:
29.34027776 + -10.83333333x + x2 = 4 + 29.34027776

Combine like terms: 4 + 29.34027776 = 33.34027776
29.34027776 + -10.83333333x + x2 = 33.34027776

Factor a perfect square on the left side:
(x + -5.416666665)(x + -5.416666665) = 33.34027776

Calculate the square root of the right side: 5.774104066

Break this problem into two subproblems by setting 
(x + -5.416666665) equal to 5.774104066 and -5.774104066.

Subproblem 1

x + -5.416666665 = 5.774104066 Simplifying x + -5.416666665 = 5.774104066 Reorder the terms: -5.416666665 + x = 5.774104066 Solving -5.416666665 + x = 5.774104066 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '5.416666665' to each side of the equation. -5.416666665 + 5.416666665 + x = 5.774104066 + 5.416666665 Combine like terms: -5.416666665 + 5.416666665 = 0.000000000 0.000000000 + x = 5.774104066 + 5.416666665 x = 5.774104066 + 5.416666665 Combine like terms: 5.774104066 + 5.416666665 = 11.190770731 x = 11.190770731 Simplifying x = 11.190770731

Subproblem 2

x + -5.416666665 = -5.774104066 Simplifying x + -5.416666665 = -5.774104066 Reorder the terms: -5.416666665 + x = -5.774104066 Solving -5.416666665 + x = -5.774104066 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '5.416666665' to each side of the equation. -5.416666665 + 5.416666665 + x = -5.774104066 + 5.416666665 Combine like terms: -5.416666665 + 5.416666665 = 0.000000000 0.000000000 + x = -5.774104066 + 5.416666665 x = -5.774104066 + 5.416666665 Combine like terms: -5.774104066 + 5.416666665 = -0.357437401 x = -0.357437401 Simplifying x = -0.357437401

Solution

The solution to the problem is based on the solutions from the subproblems. x = {11.190770731, -0.357437401}

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