3x^2+21x-16=x

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Solution for 3x^2+21x-16=x equation: 


Simplifying
3x2 + 21x + -16 = x

Reorder the terms:
-16 + 21x + 3x2 = x

Solving
-16 + 21x + 3x2 = x

Solving for variable 'x'.

Reorder the terms:
-16 + 21x + -1x + 3x2 = x + -1x

Combine like terms: 21x + -1x = 20x
-16 + 20x + 3x2 = x + -1x

Combine like terms: x + -1x = 0
-16 + 20x + 3x2 = 0

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

Divide each side by '3'.
-5.333333333 + 6.666666667x + x2 = 0

Move the constant term to the right:

Add '5.333333333' to each side of the equation.
-5.333333333 + 6.666666667x + 5.333333333 + x2 = 0 + 5.333333333

Reorder the terms:
-5.333333333 + 5.333333333 + 6.666666667x + x2 = 0 + 5.333333333

Combine like terms: -5.333333333 + 5.333333333 = 0.000000000
0.000000000 + 6.666666667x + x2 = 0 + 5.333333333
6.666666667x + x2 = 0 + 5.333333333

Combine like terms: 0 + 5.333333333 = 5.333333333
6.666666667x + x2 = 5.333333333

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 = 5.333333333 + 11.11111112

Reorder the terms:
11.11111112 + 6.666666667x + x2 = 5.333333333 + 11.11111112

Combine like terms: 5.333333333 + 11.11111112 = 16.444444453
11.11111112 + 6.666666667x + x2 = 16.444444453

Factor a perfect square on the left side:
(x + 3.333333334)(x + 3.333333334) = 16.444444453

Calculate the square root of the right side: 4.055175021

Break this problem into two subproblems by setting 
(x + 3.333333334) equal to 4.055175021 and -4.055175021.


Subproblem 1
x + 3.333333334 = 4.055175021

Simplifying
x + 3.333333334 = 4.055175021

Reorder the terms:
3.333333334 + x = 4.055175021

Solving
3.333333334 + x = 4.055175021

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 = 4.055175021 + -3.333333334

Combine like terms: 3.333333334 + -3.333333334 = 0.000000000
0.000000000 + x = 4.055175021 + -3.333333334
x = 4.055175021 + -3.333333334

Combine like terms: 4.055175021 + -3.333333334 = 0.721841687
x = 0.721841687

Simplifying
x = 0.721841687


Subproblem 2
x + 3.333333334 = -4.055175021

Simplifying
x + 3.333333334 = -4.055175021

Reorder the terms:
3.333333334 + x = -4.055175021

Solving
3.333333334 + x = -4.055175021

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 = -4.055175021 + -3.333333334

Combine like terms: 3.333333334 + -3.333333334 = 0.000000000
0.000000000 + x = -4.055175021 + -3.333333334
x = -4.055175021 + -3.333333334

Combine like terms: -4.055175021 + -3.333333334 = -7.388508355
x = -7.388508355

Simplifying
x = -7.388508355


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.721841687, -7.388508355}

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