3x^2-14x-51=0

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Solution for 3x^2-14x-51=0 equation: 


Simplifying
3x2 + -14x + -51 = 0

Reorder the terms:
-51 + -14x + 3x2 = 0

Solving
-51 + -14x + 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 + -4.666666667x + x2 = 0

Move the constant term to the right:

Add '17' to each side of the equation.
-17 + -4.666666667x + 17 + x2 = 0 + 17

Reorder the terms:
-17 + 17 + -4.666666667x + x2 = 0 + 17

Combine like terms: -17 + 17 = 0
0 + -4.666666667x + x2 = 0 + 17
-4.666666667x + x2 = 0 + 17

Combine like terms: 0 + 17 = 17
-4.666666667x + x2 = 17

The x term is -4.666666667x.  Take half its coefficient (-2.333333334).
Square it (5.444444448) and add it to both sides.

Add '5.444444448' to each side of the equation.
-4.666666667x + 5.444444448 + x2 = 17 + 5.444444448

Reorder the terms:
5.444444448 + -4.666666667x + x2 = 17 + 5.444444448

Combine like terms: 17 + 5.444444448 = 22.444444448
5.444444448 + -4.666666667x + x2 = 22.444444448

Factor a perfect square on the left side:
(x + -2.333333334)(x + -2.333333334) = 22.444444448

Calculate the square root of the right side: 4.737556802

Break this problem into two subproblems by setting 
(x + -2.333333334) equal to 4.737556802 and -4.737556802.


Subproblem 1
x + -2.333333334 = 4.737556802

Simplifying
x + -2.333333334 = 4.737556802

Reorder the terms:
-2.333333334 + x = 4.737556802

Solving
-2.333333334 + x = 4.737556802

Solving for variable 'x'.

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

Add '2.333333334' to each side of the equation.
-2.333333334 + 2.333333334 + x = 4.737556802 + 2.333333334

Combine like terms: -2.333333334 + 2.333333334 = 0.000000000
0.000000000 + x = 4.737556802 + 2.333333334
x = 4.737556802 + 2.333333334

Combine like terms: 4.737556802 + 2.333333334 = 7.070890136
x = 7.070890136

Simplifying
x = 7.070890136


Subproblem 2
x + -2.333333334 = -4.737556802

Simplifying
x + -2.333333334 = -4.737556802

Reorder the terms:
-2.333333334 + x = -4.737556802

Solving
-2.333333334 + x = -4.737556802

Solving for variable 'x'.

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

Add '2.333333334' to each side of the equation.
-2.333333334 + 2.333333334 + x = -4.737556802 + 2.333333334

Combine like terms: -2.333333334 + 2.333333334 = 0.000000000
0.000000000 + x = -4.737556802 + 2.333333334
x = -4.737556802 + 2.333333334

Combine like terms: -4.737556802 + 2.333333334 = -2.404223468
x = -2.404223468

Simplifying
x = -2.404223468


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {7.070890136, -2.404223468}

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