3x^2-38x-84=0

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Solution for 3x^2-38x-84=0 equation:


Simplifying
3x2 + -38x + -84 = 0

Reorder the terms:
-84 + -38x + 3x2 = 0

Solving
-84 + -38x + 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'.
-28 + -12.66666667x + x2 = 0

Move the constant term to the right:

Add '28' to each side of the equation.
-28 + -12.66666667x + 28 + x2 = 0 + 28

Reorder the terms:
-28 + 28 + -12.66666667x + x2 = 0 + 28

Combine like terms: -28 + 28 = 0
0 + -12.66666667x + x2 = 0 + 28
-12.66666667x + x2 = 0 + 28

Combine like terms: 0 + 28 = 28
-12.66666667x + x2 = 28

The x term is -12.66666667x.  Take half its coefficient (-6.333333335).
Square it (40.11111113) and add it to both sides.

Add '40.11111113' to each side of the equation.
-12.66666667x + 40.11111113 + x2 = 28 + 40.11111113

Reorder the terms:
40.11111113 + -12.66666667x + x2 = 28 + 40.11111113

Combine like terms: 28 + 40.11111113 = 68.11111113
40.11111113 + -12.66666667x + x2 = 68.11111113

Factor a perfect square on the left side:
(x + -6.333333335)(x + -6.333333335) = 68.11111113

Calculate the square root of the right side: 8.252945603

Break this problem into two subproblems by setting 
(x + -6.333333335) equal to 8.252945603 and -8.252945603.

Subproblem 1

x + -6.333333335 = 8.252945603 Simplifying x + -6.333333335 = 8.252945603 Reorder the terms: -6.333333335 + x = 8.252945603 Solving -6.333333335 + x = 8.252945603 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '6.333333335' to each side of the equation. -6.333333335 + 6.333333335 + x = 8.252945603 + 6.333333335 Combine like terms: -6.333333335 + 6.333333335 = 0.000000000 0.000000000 + x = 8.252945603 + 6.333333335 x = 8.252945603 + 6.333333335 Combine like terms: 8.252945603 + 6.333333335 = 14.586278938 x = 14.586278938 Simplifying x = 14.586278938

Subproblem 2

x + -6.333333335 = -8.252945603 Simplifying x + -6.333333335 = -8.252945603 Reorder the terms: -6.333333335 + x = -8.252945603 Solving -6.333333335 + x = -8.252945603 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '6.333333335' to each side of the equation. -6.333333335 + 6.333333335 + x = -8.252945603 + 6.333333335 Combine like terms: -6.333333335 + 6.333333335 = 0.000000000 0.000000000 + x = -8.252945603 + 6.333333335 x = -8.252945603 + 6.333333335 Combine like terms: -8.252945603 + 6.333333335 = -1.919612268 x = -1.919612268 Simplifying x = -1.919612268

Solution

The solution to the problem is based on the solutions from the subproblems. x = {14.586278938, -1.919612268}

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