3x^2-13x-51=0

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Solution for 3x^2-13x-51=0 equation: 


Simplifying
3x2 + -13x + -51 = 0

Reorder the terms:
-51 + -13x + 3x2 = 0

Solving
-51 + -13x + 3x2 = 0

Solving for variable 'x'.

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

Divide each side by '3'.
-17 + -4.333333333x + x2 = 0

Move the constant term to the right:

Add '17' to each side of the equation.
-17 + -4.333333333x + 17 + x2 = 0 + 17

Reorder the terms:
-17 + 17 + -4.333333333x + x2 = 0 + 17

Combine like terms: -17 + 17 = 0
0 + -4.333333333x + x2 = 0 + 17
-4.333333333x + x2 = 0 + 17

Combine like terms: 0 + 17 = 17
-4.333333333x + x2 = 17

The x term is -4.333333333x.  Take half its coefficient (-2.166666667).
Square it (4.694444446) and add it to both sides.

Add '4.694444446' to each side of the equation.
-4.333333333x + 4.694444446 + x2 = 17 + 4.694444446

Reorder the terms:
4.694444446 + -4.333333333x + x2 = 17 + 4.694444446

Combine like terms: 17 + 4.694444446 = 21.694444446
4.694444446 + -4.333333333x + x2 = 21.694444446

Factor a perfect square on the left side:
(x + -2.166666667)(x + -2.166666667) = 21.694444446

Calculate the square root of the right side: 4.657729538

Break this problem into two subproblems by setting 
(x + -2.166666667) equal to 4.657729538 and -4.657729538.


Subproblem 1
x + -2.166666667 = 4.657729538

Simplifying
x + -2.166666667 = 4.657729538

Reorder the terms:
-2.166666667 + x = 4.657729538

Solving
-2.166666667 + x = 4.657729538

Solving for variable 'x'.

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

Add '2.166666667' to each side of the equation.
-2.166666667 + 2.166666667 + x = 4.657729538 + 2.166666667

Combine like terms: -2.166666667 + 2.166666667 = 0.000000000
0.000000000 + x = 4.657729538 + 2.166666667
x = 4.657729538 + 2.166666667

Combine like terms: 4.657729538 + 2.166666667 = 6.824396205
x = 6.824396205

Simplifying
x = 6.824396205


Subproblem 2
x + -2.166666667 = -4.657729538

Simplifying
x + -2.166666667 = -4.657729538

Reorder the terms:
-2.166666667 + x = -4.657729538

Solving
-2.166666667 + x = -4.657729538

Solving for variable 'x'.

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

Add '2.166666667' to each side of the equation.
-2.166666667 + 2.166666667 + x = -4.657729538 + 2.166666667

Combine like terms: -2.166666667 + 2.166666667 = 0.000000000
0.000000000 + x = -4.657729538 + 2.166666667
x = -4.657729538 + 2.166666667

Combine like terms: -4.657729538 + 2.166666667 = -2.491062871
x = -2.491062871

Simplifying
x = -2.491062871


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {6.824396205, -2.491062871}

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