(3x^2)-5x-7=0

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Solution for (3x^2)-5x-7=0 equation: 


Simplifying
(3x2) + -5x + -7 = 0

Reorder the terms:
-7 + -5x + (3x2) = 0

Solving
-7 + -5x + (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'.
-2.333333333 + -1.666666667x + x2 = 0

Move the constant term to the right:

Add '2.333333333' to each side of the equation.
-2.333333333 + -1.666666667x + 2.333333333 + x2 = 0 + 2.333333333

Reorder the terms:
-2.333333333 + 2.333333333 + -1.666666667x + x2 = 0 + 2.333333333

Combine like terms: -2.333333333 + 2.333333333 = 0.000000000
0.000000000 + -1.666666667x + x2 = 0 + 2.333333333
-1.666666667x + x2 = 0 + 2.333333333

Combine like terms: 0 + 2.333333333 = 2.333333333
-1.666666667x + x2 = 2.333333333

The x term is -1.666666667x.  Take half its coefficient (-0.8333333335).
Square it (0.6944444447) and add it to both sides.

Add '0.6944444447' to each side of the equation.
-1.666666667x + 0.6944444447 + x2 = 2.333333333 + 0.6944444447

Reorder the terms:
0.6944444447 + -1.666666667x + x2 = 2.333333333 + 0.6944444447

Combine like terms: 2.333333333 + 0.6944444447 = 3.0277777777
0.6944444447 + -1.666666667x + x2 = 3.0277777777

Factor a perfect square on the left side:
((x) + -0.8333333335)((x) + -0.8333333335) = 3.0277777777

Calculate the square root of the right side: 1.740051085

Break this problem into two subproblems by setting 
((x) + -0.8333333335) equal to 1.740051085 and -1.740051085.


Subproblem 1
(x) + -0.8333333335 = 1.740051085

Simplifying
(x) + -0.8333333335 = 1.740051085
x + -0.8333333335 = 1.740051085

Reorder the terms:
-0.8333333335 + x = 1.740051085

Solving
-0.8333333335 + x = 1.740051085

Solving for variable 'x'.

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

Add '0.8333333335' to each side of the equation.
-0.8333333335 + 0.8333333335 + x = 1.740051085 + 0.8333333335

Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000
0.0000000000 + x = 1.740051085 + 0.8333333335
x = 1.740051085 + 0.8333333335

Combine like terms: 1.740051085 + 0.8333333335 = 2.5733844185
x = 2.5733844185

Simplifying
x = 2.5733844185


Subproblem 2
(x) + -0.8333333335 = -1.740051085

Simplifying
(x) + -0.8333333335 = -1.740051085
x + -0.8333333335 = -1.740051085

Reorder the terms:
-0.8333333335 + x = -1.740051085

Solving
-0.8333333335 + x = -1.740051085

Solving for variable 'x'.

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

Add '0.8333333335' to each side of the equation.
-0.8333333335 + 0.8333333335 + x = -1.740051085 + 0.8333333335

Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000
0.0000000000 + x = -1.740051085 + 0.8333333335
x = -1.740051085 + 0.8333333335

Combine like terms: -1.740051085 + 0.8333333335 = -0.9067177515
x = -0.9067177515

Simplifying
x = -0.9067177515


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.5733844185, -0.9067177515}

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