3x^2+5x+6=38

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Solution for 3x^2+5x+6=38 equation:


Simplifying
3x2 + 5x + 6 = 38

Reorder the terms:
6 + 5x + 3x2 = 38

Solving
6 + 5x + 3x2 = 38

Solving for variable 'x'.

Reorder the terms:
6 + -38 + 5x + 3x2 = 38 + -38

Combine like terms: 6 + -38 = -32
-32 + 5x + 3x2 = 38 + -38

Combine like terms: 38 + -38 = 0
-32 + 5x + 3x2 = 0

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

Divide each side by '3'.
-10.66666667 + 1.666666667x + x2 = 0

Move the constant term to the right:

Add '10.66666667' to each side of the equation.
-10.66666667 + 1.666666667x + 10.66666667 + x2 = 0 + 10.66666667

Reorder the terms:
-10.66666667 + 10.66666667 + 1.666666667x + x2 = 0 + 10.66666667

Combine like terms: -10.66666667 + 10.66666667 = 0.00000000
0.00000000 + 1.666666667x + x2 = 0 + 10.66666667
1.666666667x + x2 = 0 + 10.66666667

Combine like terms: 0 + 10.66666667 = 10.66666667
1.666666667x + x2 = 10.66666667

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 = 10.66666667 + 0.6944444447

Reorder the terms:
0.6944444447 + 1.666666667x + x2 = 10.66666667 + 0.6944444447

Combine like terms: 10.66666667 + 0.6944444447 = 11.3611111147
0.6944444447 + 1.666666667x + x2 = 11.3611111147

Factor a perfect square on the left side:
(x + 0.8333333335)(x + 0.8333333335) = 11.3611111147

Calculate the square root of the right side: 3.370624737

Break this problem into two subproblems by setting 
(x + 0.8333333335) equal to 3.370624737 and -3.370624737.

Subproblem 1

x + 0.8333333335 = 3.370624737 Simplifying x + 0.8333333335 = 3.370624737 Reorder the terms: 0.8333333335 + x = 3.370624737 Solving 0.8333333335 + x = 3.370624737 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 = 3.370624737 + -0.8333333335 Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000 0.0000000000 + x = 3.370624737 + -0.8333333335 x = 3.370624737 + -0.8333333335 Combine like terms: 3.370624737 + -0.8333333335 = 2.5372914035 x = 2.5372914035 Simplifying x = 2.5372914035

Subproblem 2

x + 0.8333333335 = -3.370624737 Simplifying x + 0.8333333335 = -3.370624737 Reorder the terms: 0.8333333335 + x = -3.370624737 Solving 0.8333333335 + x = -3.370624737 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 = -3.370624737 + -0.8333333335 Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000 0.0000000000 + x = -3.370624737 + -0.8333333335 x = -3.370624737 + -0.8333333335 Combine like terms: -3.370624737 + -0.8333333335 = -4.2039580705 x = -4.2039580705 Simplifying x = -4.2039580705

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.5372914035, -4.2039580705}

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