4x^2-6x-22=0

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Solution for 4x^2-6x-22=0 equation: 


Simplifying
4x2 + -6x + -22 = 0

Reorder the terms:
-22 + -6x + 4x2 = 0

Solving
-22 + -6x + 4x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '2'.
2(-11 + -3x + 2x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-11 + -3x + 2x2)' equal to zero and attempt to solve:

Simplifying
-11 + -3x + 2x2 = 0

Solving
-11 + -3x + 2x2 = 0

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

Divide each side by '2'.
-5.5 + -1.5x + x2 = 0

Move the constant term to the right:

Add '5.5' to each side of the equation.
-5.5 + -1.5x + 5.5 + x2 = 0 + 5.5

Reorder the terms:
-5.5 + 5.5 + -1.5x + x2 = 0 + 5.5

Combine like terms: -5.5 + 5.5 = 0.0
0.0 + -1.5x + x2 = 0 + 5.5
-1.5x + x2 = 0 + 5.5

Combine like terms: 0 + 5.5 = 5.5
-1.5x + x2 = 5.5

The x term is -1.5x.  Take half its coefficient (-0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
-1.5x + 0.5625 + x2 = 5.5 + 0.5625

Reorder the terms:
0.5625 + -1.5x + x2 = 5.5 + 0.5625

Combine like terms: 5.5 + 0.5625 = 6.0625
0.5625 + -1.5x + x2 = 6.0625

Factor a perfect square on the left side:
(x + -0.75)(x + -0.75) = 6.0625

Calculate the square root of the right side: 2.46221445

Break this problem into two subproblems by setting 
(x + -0.75) equal to 2.46221445 and -2.46221445.


Subproblem 1
x + -0.75 = 2.46221445

Simplifying
x + -0.75 = 2.46221445

Reorder the terms:
-0.75 + x = 2.46221445

Solving
-0.75 + x = 2.46221445

Solving for variable 'x'.

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

Add '0.75' to each side of the equation.
-0.75 + 0.75 + x = 2.46221445 + 0.75

Combine like terms: -0.75 + 0.75 = 0.00
0.00 + x = 2.46221445 + 0.75
x = 2.46221445 + 0.75

Combine like terms: 2.46221445 + 0.75 = 3.21221445
x = 3.21221445

Simplifying
x = 3.21221445


Subproblem 2
x + -0.75 = -2.46221445

Simplifying
x + -0.75 = -2.46221445

Reorder the terms:
-0.75 + x = -2.46221445

Solving
-0.75 + x = -2.46221445

Solving for variable 'x'.

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

Add '0.75' to each side of the equation.
-0.75 + 0.75 + x = -2.46221445 + 0.75

Combine like terms: -0.75 + 0.75 = 0.00
0.00 + x = -2.46221445 + 0.75
x = -2.46221445 + 0.75

Combine like terms: -2.46221445 + 0.75 = -1.71221445
x = -1.71221445

Simplifying
x = -1.71221445


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.21221445, -1.71221445}
Solution
x = {3.21221445, -1.71221445}

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