-4x^2+22x-14=0

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Solution for -4x^2+22x-14=0 equation: 


Simplifying
-4x2 + 22x + -14 = 0

Reorder the terms:
-14 + 22x + -4x2 = 0

Solving
-14 + 22x + -4x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '2'.
2(-7 + 11x + -2x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-7 + 11x + -2x2)' equal to zero and attempt to solve:

Simplifying
-7 + 11x + -2x2 = 0

Solving
-7 + 11x + -2x2 = 0

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

Divide each side by '-2'.
3.5 + -5.5x + x2 = 0

Move the constant term to the right:

Add '-3.5' to each side of the equation.
3.5 + -5.5x + -3.5 + x2 = 0 + -3.5

Reorder the terms:
3.5 + -3.5 + -5.5x + x2 = 0 + -3.5

Combine like terms: 3.5 + -3.5 = 0.0
0.0 + -5.5x + x2 = 0 + -3.5
-5.5x + x2 = 0 + -3.5

Combine like terms: 0 + -3.5 = -3.5
-5.5x + x2 = -3.5

The x term is -5.5x.  Take half its coefficient (-2.75).
Square it (7.5625) and add it to both sides.

Add '7.5625' to each side of the equation.
-5.5x + 7.5625 + x2 = -3.5 + 7.5625

Reorder the terms:
7.5625 + -5.5x + x2 = -3.5 + 7.5625

Combine like terms: -3.5 + 7.5625 = 4.0625
7.5625 + -5.5x + x2 = 4.0625

Factor a perfect square on the left side:
(x + -2.75)(x + -2.75) = 4.0625

Calculate the square root of the right side: 2.015564437

Break this problem into two subproblems by setting 
(x + -2.75) equal to 2.015564437 and -2.015564437.


Subproblem 1
x + -2.75 = 2.015564437

Simplifying
x + -2.75 = 2.015564437

Reorder the terms:
-2.75 + x = 2.015564437

Solving
-2.75 + x = 2.015564437

Solving for variable 'x'.

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

Add '2.75' to each side of the equation.
-2.75 + 2.75 + x = 2.015564437 + 2.75

Combine like terms: -2.75 + 2.75 = 0.00
0.00 + x = 2.015564437 + 2.75
x = 2.015564437 + 2.75

Combine like terms: 2.015564437 + 2.75 = 4.765564437
x = 4.765564437

Simplifying
x = 4.765564437


Subproblem 2
x + -2.75 = -2.015564437

Simplifying
x + -2.75 = -2.015564437

Reorder the terms:
-2.75 + x = -2.015564437

Solving
-2.75 + x = -2.015564437

Solving for variable 'x'.

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

Add '2.75' to each side of the equation.
-2.75 + 2.75 + x = -2.015564437 + 2.75

Combine like terms: -2.75 + 2.75 = 0.00
0.00 + x = -2.015564437 + 2.75
x = -2.015564437 + 2.75

Combine like terms: -2.015564437 + 2.75 = 0.734435563
x = 0.734435563

Simplifying
x = 0.734435563


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {4.765564437, 0.734435563}
Solution
x = {4.765564437, 0.734435563}

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