3x^2-38x+114=0

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Solution for 3x^2-38x+114=0 equation:


Simplifying
3x2 + -38x + 114 = 0

Reorder the terms:
114 + -38x + 3x2 = 0

Solving
114 + -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'.
38 + -12.66666667x + x2 = 0

Move the constant term to the right:

Add '-38' to each side of the equation.
38 + -12.66666667x + -38 + x2 = 0 + -38

Reorder the terms:
38 + -38 + -12.66666667x + x2 = 0 + -38

Combine like terms: 38 + -38 = 0
0 + -12.66666667x + x2 = 0 + -38
-12.66666667x + x2 = 0 + -38

Combine like terms: 0 + -38 = -38
-12.66666667x + x2 = -38

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 = -38 + 40.11111113

Reorder the terms:
40.11111113 + -12.66666667x + x2 = -38 + 40.11111113

Combine like terms: -38 + 40.11111113 = 2.11111113
40.11111113 + -12.66666667x + x2 = 2.11111113

Factor a perfect square on the left side:
(x + -6.333333335)(x + -6.333333335) = 2.11111113

Calculate the square root of the right side: 1.452966321

Break this problem into two subproblems by setting 
(x + -6.333333335) equal to 1.452966321 and -1.452966321.

Subproblem 1

x + -6.333333335 = 1.452966321 Simplifying x + -6.333333335 = 1.452966321 Reorder the terms: -6.333333335 + x = 1.452966321 Solving -6.333333335 + x = 1.452966321 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 = 1.452966321 + 6.333333335 Combine like terms: -6.333333335 + 6.333333335 = 0.000000000 0.000000000 + x = 1.452966321 + 6.333333335 x = 1.452966321 + 6.333333335 Combine like terms: 1.452966321 + 6.333333335 = 7.786299656 x = 7.786299656 Simplifying x = 7.786299656

Subproblem 2

x + -6.333333335 = -1.452966321 Simplifying x + -6.333333335 = -1.452966321 Reorder the terms: -6.333333335 + x = -1.452966321 Solving -6.333333335 + x = -1.452966321 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 = -1.452966321 + 6.333333335 Combine like terms: -6.333333335 + 6.333333335 = 0.000000000 0.000000000 + x = -1.452966321 + 6.333333335 x = -1.452966321 + 6.333333335 Combine like terms: -1.452966321 + 6.333333335 = 4.880367014 x = 4.880367014 Simplifying x = 4.880367014

Solution

The solution to the problem is based on the solutions from the subproblems. x = {7.786299656, 4.880367014}

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