3x^2+22x-6=0

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Solution for 3x^2+22x-6=0 equation:


Simplifying
3x2 + 22x + -6 = 0

Reorder the terms:
-6 + 22x + 3x2 = 0

Solving
-6 + 22x + 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 + 7.333333333x + x2 = 0

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + 7.333333333x + 2 + x2 = 0 + 2

Reorder the terms:
-2 + 2 + 7.333333333x + x2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + 7.333333333x + x2 = 0 + 2
7.333333333x + x2 = 0 + 2

Combine like terms: 0 + 2 = 2
7.333333333x + x2 = 2

The x term is 7.333333333x.  Take half its coefficient (3.666666667).
Square it (13.44444445) and add it to both sides.

Add '13.44444445' to each side of the equation.
7.333333333x + 13.44444445 + x2 = 2 + 13.44444445

Reorder the terms:
13.44444445 + 7.333333333x + x2 = 2 + 13.44444445

Combine like terms: 2 + 13.44444445 = 15.44444445
13.44444445 + 7.333333333x + x2 = 15.44444445

Factor a perfect square on the left side:
(x + 3.666666667)(x + 3.666666667) = 15.44444445

Calculate the square root of the right side: 3.929942042

Break this problem into two subproblems by setting 
(x + 3.666666667) equal to 3.929942042 and -3.929942042.

Subproblem 1

x + 3.666666667 = 3.929942042 Simplifying x + 3.666666667 = 3.929942042 Reorder the terms: 3.666666667 + x = 3.929942042 Solving 3.666666667 + x = 3.929942042 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.666666667' to each side of the equation. 3.666666667 + -3.666666667 + x = 3.929942042 + -3.666666667 Combine like terms: 3.666666667 + -3.666666667 = 0.000000000 0.000000000 + x = 3.929942042 + -3.666666667 x = 3.929942042 + -3.666666667 Combine like terms: 3.929942042 + -3.666666667 = 0.263275375 x = 0.263275375 Simplifying x = 0.263275375

Subproblem 2

x + 3.666666667 = -3.929942042 Simplifying x + 3.666666667 = -3.929942042 Reorder the terms: 3.666666667 + x = -3.929942042 Solving 3.666666667 + x = -3.929942042 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.666666667' to each side of the equation. 3.666666667 + -3.666666667 + x = -3.929942042 + -3.666666667 Combine like terms: 3.666666667 + -3.666666667 = 0.000000000 0.000000000 + x = -3.929942042 + -3.666666667 x = -3.929942042 + -3.666666667 Combine like terms: -3.929942042 + -3.666666667 = -7.596608709 x = -7.596608709 Simplifying x = -7.596608709

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.263275375, -7.596608709}

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