-4x^2+23x-14=0

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Solution for -4x^2+23x-14=0 equation: 


Simplifying
-4x2 + 23x + -14 = 0

Reorder the terms:
-14 + 23x + -4x2 = 0

Solving
-14 + 23x + -4x2 = 0

Solving for variable 'x'.

Begin completing the square.  Divide all terms by
-4 the coefficient of the squared term: 

Divide each side by '-4'.
3.5 + -5.75x + x2 = 0

Move the constant term to the right:

Add '-3.5' to each side of the equation.
3.5 + -5.75x + -3.5 + x2 = 0 + -3.5

Reorder the terms:
3.5 + -3.5 + -5.75x + x2 = 0 + -3.5

Combine like terms: 3.5 + -3.5 = 0.0
0.0 + -5.75x + x2 = 0 + -3.5
-5.75x + x2 = 0 + -3.5

Combine like terms: 0 + -3.5 = -3.5
-5.75x + x2 = -3.5

The x term is -5.75x.  Take half its coefficient (-2.875).
Square it (8.265625) and add it to both sides.

Add '8.265625' to each side of the equation.
-5.75x + 8.265625 + x2 = -3.5 + 8.265625

Reorder the terms:
8.265625 + -5.75x + x2 = -3.5 + 8.265625

Combine like terms: -3.5 + 8.265625 = 4.765625
8.265625 + -5.75x + x2 = 4.765625

Factor a perfect square on the left side:
(x + -2.875)(x + -2.875) = 4.765625

Calculate the square root of the right side: 2.18303115

Break this problem into two subproblems by setting 
(x + -2.875) equal to 2.18303115 and -2.18303115.


Subproblem 1
x + -2.875 = 2.18303115

Simplifying
x + -2.875 = 2.18303115

Reorder the terms:
-2.875 + x = 2.18303115

Solving
-2.875 + x = 2.18303115

Solving for variable 'x'.

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

Add '2.875' to each side of the equation.
-2.875 + 2.875 + x = 2.18303115 + 2.875

Combine like terms: -2.875 + 2.875 = 0.000
0.000 + x = 2.18303115 + 2.875
x = 2.18303115 + 2.875

Combine like terms: 2.18303115 + 2.875 = 5.05803115
x = 5.05803115

Simplifying
x = 5.05803115


Subproblem 2
x + -2.875 = -2.18303115

Simplifying
x + -2.875 = -2.18303115

Reorder the terms:
-2.875 + x = -2.18303115

Solving
-2.875 + x = -2.18303115

Solving for variable 'x'.

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

Add '2.875' to each side of the equation.
-2.875 + 2.875 + x = -2.18303115 + 2.875

Combine like terms: -2.875 + 2.875 = 0.000
0.000 + x = -2.18303115 + 2.875
x = -2.18303115 + 2.875

Combine like terms: -2.18303115 + 2.875 = 0.69196885
x = 0.69196885

Simplifying
x = 0.69196885


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {5.05803115, 0.69196885}

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