x(3x+21)=23

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Solution for x(3x+21)=23 equation: 


Simplifying
x(3x + 21) = 23

Reorder the terms:
x(21 + 3x) = 23
(21 * x + 3x * x) = 23
(21x + 3x2) = 23

Solving
21x + 3x2 = 23

Solving for variable 'x'.

Reorder the terms:
-23 + 21x + 3x2 = 23 + -23

Combine like terms: 23 + -23 = 0
-23 + 21x + 3x2 = 0

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

Divide each side by '3'.
-7.666666667 + 7x + x2 = 0

Move the constant term to the right:

Add '7.666666667' to each side of the equation.
-7.666666667 + 7x + 7.666666667 + x2 = 0 + 7.666666667

Reorder the terms:
-7.666666667 + 7.666666667 + 7x + x2 = 0 + 7.666666667

Combine like terms: -7.666666667 + 7.666666667 = 0.000000000
0.000000000 + 7x + x2 = 0 + 7.666666667
7x + x2 = 0 + 7.666666667

Combine like terms: 0 + 7.666666667 = 7.666666667
7x + x2 = 7.666666667

The x term is 7x.  Take half its coefficient (3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
7x + 12.25 + x2 = 7.666666667 + 12.25

Reorder the terms:
12.25 + 7x + x2 = 7.666666667 + 12.25

Combine like terms: 7.666666667 + 12.25 = 19.916666667
12.25 + 7x + x2 = 19.916666667

Factor a perfect square on the left side:
(x + 3.5)(x + 3.5) = 19.916666667

Calculate the square root of the right side: 4.46280928

Break this problem into two subproblems by setting 
(x + 3.5) equal to 4.46280928 and -4.46280928.


Subproblem 1
x + 3.5 = 4.46280928

Simplifying
x + 3.5 = 4.46280928

Reorder the terms:
3.5 + x = 4.46280928

Solving
3.5 + x = 4.46280928

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-3.5' to each side of the equation.
3.5 + -3.5 + x = 4.46280928 + -3.5

Combine like terms: 3.5 + -3.5 = 0.0
0.0 + x = 4.46280928 + -3.5
x = 4.46280928 + -3.5

Combine like terms: 4.46280928 + -3.5 = 0.96280928
x = 0.96280928

Simplifying
x = 0.96280928


Subproblem 2
x + 3.5 = -4.46280928

Simplifying
x + 3.5 = -4.46280928

Reorder the terms:
3.5 + x = -4.46280928

Solving
3.5 + x = -4.46280928

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-3.5' to each side of the equation.
3.5 + -3.5 + x = -4.46280928 + -3.5

Combine like terms: 3.5 + -3.5 = 0.0
0.0 + x = -4.46280928 + -3.5
x = -4.46280928 + -3.5

Combine like terms: -4.46280928 + -3.5 = -7.96280928
x = -7.96280928

Simplifying
x = -7.96280928


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.96280928, -7.96280928}

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