3x^2-21x-51=0

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Solution for 3x^2-21x-51=0 equation: 


Simplifying
3x2 + -21x + -51 = 0

Reorder the terms:
-51 + -21x + 3x2 = 0

Solving
-51 + -21x + 3x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '3'.
3(-17 + -7x + x2) = 0

Ignore the factor 3.

Subproblem 1
Set the factor '(-17 + -7x + x2)' equal to zero and attempt to solve:

Simplifying
-17 + -7x + x2 = 0

Solving
-17 + -7x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '17' to each side of the equation.
-17 + -7x + 17 + x2 = 0 + 17

Reorder the terms:
-17 + 17 + -7x + x2 = 0 + 17

Combine like terms: -17 + 17 = 0
0 + -7x + x2 = 0 + 17
-7x + x2 = 0 + 17

Combine like terms: 0 + 17 = 17
-7x + x2 = 17

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 = 17 + 12.25

Reorder the terms:
12.25 + -7x + x2 = 17 + 12.25

Combine like terms: 17 + 12.25 = 29.25
12.25 + -7x + x2 = 29.25

Factor a perfect square on the left side:
(x + -3.5)(x + -3.5) = 29.25

Calculate the square root of the right side: 5.408326913

Break this problem into two subproblems by setting 
(x + -3.5) equal to 5.408326913 and -5.408326913.


Subproblem 1
x + -3.5 = 5.408326913

Simplifying
x + -3.5 = 5.408326913

Reorder the terms:
-3.5 + x = 5.408326913

Solving
-3.5 + x = 5.408326913

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 = 5.408326913 + 3.5

Combine like terms: -3.5 + 3.5 = 0.0
0.0 + x = 5.408326913 + 3.5
x = 5.408326913 + 3.5

Combine like terms: 5.408326913 + 3.5 = 8.908326913
x = 8.908326913

Simplifying
x = 8.908326913


Subproblem 2
x + -3.5 = -5.408326913

Simplifying
x + -3.5 = -5.408326913

Reorder the terms:
-3.5 + x = -5.408326913

Solving
-3.5 + x = -5.408326913

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 = -5.408326913 + 3.5

Combine like terms: -3.5 + 3.5 = 0.0
0.0 + x = -5.408326913 + 3.5
x = -5.408326913 + 3.5

Combine like terms: -5.408326913 + 3.5 = -1.908326913
x = -1.908326913

Simplifying
x = -1.908326913


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {8.908326913, -1.908326913}
Solution
x = {8.908326913, -1.908326913}

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