6x^2+22x-5=0

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Solution for 6x^2+22x-5=0 equation: 


Simplifying
6x2 + 22x + -5 = 0

Reorder the terms:
-5 + 22x + 6x2 = 0

Solving
-5 + 22x + 6x2 = 0

Solving for variable 'x'.

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

Divide each side by '6'.
-0.8333333333 + 3.666666667x + x2 = 0

Move the constant term to the right:

Add '0.8333333333' to each side of the equation.
-0.8333333333 + 3.666666667x + 0.8333333333 + x2 = 0 + 0.8333333333

Reorder the terms:
-0.8333333333 + 0.8333333333 + 3.666666667x + x2 = 0 + 0.8333333333

Combine like terms: -0.8333333333 + 0.8333333333 = 0.0000000000
0.0000000000 + 3.666666667x + x2 = 0 + 0.8333333333
3.666666667x + x2 = 0 + 0.8333333333

Combine like terms: 0 + 0.8333333333 = 0.8333333333
3.666666667x + x2 = 0.8333333333

The x term is 3.666666667x.  Take half its coefficient (1.833333334).
Square it (3.361111114) and add it to both sides.

Add '3.361111114' to each side of the equation.
3.666666667x + 3.361111114 + x2 = 0.8333333333 + 3.361111114

Reorder the terms:
3.361111114 + 3.666666667x + x2 = 0.8333333333 + 3.361111114

Combine like terms: 0.8333333333 + 3.361111114 = 4.1944444473
3.361111114 + 3.666666667x + x2 = 4.1944444473

Factor a perfect square on the left side:
(x + 1.833333334)(x + 1.833333334) = 4.1944444473

Calculate the square root of the right side: 2.048034289

Break this problem into two subproblems by setting 
(x + 1.833333334) equal to 2.048034289 and -2.048034289.


Subproblem 1
x + 1.833333334 = 2.048034289

Simplifying
x + 1.833333334 = 2.048034289

Reorder the terms:
1.833333334 + x = 2.048034289

Solving
1.833333334 + x = 2.048034289

Solving for variable 'x'.

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

Add '-1.833333334' to each side of the equation.
1.833333334 + -1.833333334 + x = 2.048034289 + -1.833333334

Combine like terms: 1.833333334 + -1.833333334 = 0.000000000
0.000000000 + x = 2.048034289 + -1.833333334
x = 2.048034289 + -1.833333334

Combine like terms: 2.048034289 + -1.833333334 = 0.214700955
x = 0.214700955

Simplifying
x = 0.214700955


Subproblem 2
x + 1.833333334 = -2.048034289

Simplifying
x + 1.833333334 = -2.048034289

Reorder the terms:
1.833333334 + x = -2.048034289

Solving
1.833333334 + x = -2.048034289

Solving for variable 'x'.

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

Add '-1.833333334' to each side of the equation.
1.833333334 + -1.833333334 + x = -2.048034289 + -1.833333334

Combine like terms: 1.833333334 + -1.833333334 = 0.000000000
0.000000000 + x = -2.048034289 + -1.833333334
x = -2.048034289 + -1.833333334

Combine like terms: -2.048034289 + -1.833333334 = -3.881367623
x = -3.881367623

Simplifying
x = -3.881367623


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.214700955, -3.881367623}

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