-4.905x^2+23.23x-11=0

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Solution for -4.905x^2+23.23x-11=0 equation:


Simplifying
-4.905x2 + 23.23x + -11 = 0

Reorder the terms:
-11 + 23.23x + -4.905x2 = 0

Solving
-11 + 23.23x + -4.905x2 = 0

Solving for variable 'x'.

Begin completing the square.  Divide all terms by
-4.905 the coefficient of the squared term: 

Divide each side by '-4.905'.
2.242609582 + -4.73598369x + x2 = 0

Move the constant term to the right:

Add '-2.242609582' to each side of the equation.
2.242609582 + -4.73598369x + -2.242609582 + x2 = 0 + -2.242609582

Reorder the terms:
2.242609582 + -2.242609582 + -4.73598369x + x2 = 0 + -2.242609582

Combine like terms: 2.242609582 + -2.242609582 = 0.000000000
0.000000000 + -4.73598369x + x2 = 0 + -2.242609582
-4.73598369x + x2 = 0 + -2.242609582

Combine like terms: 0 + -2.242609582 = -2.242609582
-4.73598369x + x2 = -2.242609582

The x term is -4.73598369x.  Take half its coefficient (-2.367991845).
Square it (5.607385378) and add it to both sides.

Add '5.607385378' to each side of the equation.
-4.73598369x + 5.607385378 + x2 = -2.242609582 + 5.607385378

Reorder the terms:
5.607385378 + -4.73598369x + x2 = -2.242609582 + 5.607385378

Combine like terms: -2.242609582 + 5.607385378 = 3.364775796
5.607385378 + -4.73598369x + x2 = 3.364775796

Factor a perfect square on the left side:
(x + -2.367991845)(x + -2.367991845) = 3.364775796

Calculate the square root of the right side: 1.834332521

Break this problem into two subproblems by setting 
(x + -2.367991845) equal to 1.834332521 and -1.834332521.


Subproblem 1
x + -2.367991845 = 1.834332521

Simplifying
x + -2.367991845 = 1.834332521

Reorder the terms:
-2.367991845 + x = 1.834332521

Solving
-2.367991845 + x = 1.834332521

Solving for variable 'x'.

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

Add '2.367991845' to each side of the equation.
-2.367991845 + 2.367991845 + x = 1.834332521 + 2.367991845

Combine like terms: -2.367991845 + 2.367991845 = 0.000000000
0.000000000 + x = 1.834332521 + 2.367991845
x = 1.834332521 + 2.367991845

Combine like terms: 1.834332521 + 2.367991845 = 4.202324366
x = 4.202324366

Simplifying
x = 4.202324366


Subproblem 2
x + -2.367991845 = -1.834332521

Simplifying
x + -2.367991845 = -1.834332521

Reorder the terms:
-2.367991845 + x = -1.834332521

Solving
-2.367991845 + x = -1.834332521

Solving for variable 'x'.

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

Add '2.367991845' to each side of the equation.
-2.367991845 + 2.367991845 + x = -1.834332521 + 2.367991845

Combine like terms: -2.367991845 + 2.367991845 = 0.000000000
0.000000000 + x = -1.834332521 + 2.367991845
x = -1.834332521 + 2.367991845

Combine like terms: -1.834332521 + 2.367991845 = 0.533659324
x = 0.533659324

Simplifying
x = 0.533659324


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {4.202324366, 0.533659324}

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