24x^2-23x-5=0

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Solution for 24x^2-23x-5=0 equation: 


Simplifying
24x2 + -23x + -5 = 0

Reorder the terms:
-5 + -23x + 24x2 = 0

Solving
-5 + -23x + 24x2 = 0

Solving for variable 'x'.

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

Divide each side by '24'.
-0.2083333333 + -0.9583333333x + x2 = 0

Move the constant term to the right:

Add '0.2083333333' to each side of the equation.
-0.2083333333 + -0.9583333333x + 0.2083333333 + x2 = 0 + 0.2083333333

Reorder the terms:
-0.2083333333 + 0.2083333333 + -0.9583333333x + x2 = 0 + 0.2083333333

Combine like terms: -0.2083333333 + 0.2083333333 = 0.0000000000
0.0000000000 + -0.9583333333x + x2 = 0 + 0.2083333333
-0.9583333333x + x2 = 0 + 0.2083333333

Combine like terms: 0 + 0.2083333333 = 0.2083333333
-0.9583333333x + x2 = 0.2083333333

The x term is -0.9583333333x.  Take half its coefficient (-0.4791666667).
Square it (0.2296006945) and add it to both sides.

Add '0.2296006945' to each side of the equation.
-0.9583333333x + 0.2296006945 + x2 = 0.2083333333 + 0.2296006945

Reorder the terms:
0.2296006945 + -0.9583333333x + x2 = 0.2083333333 + 0.2296006945

Combine like terms: 0.2083333333 + 0.2296006945 = 0.4379340278
0.2296006945 + -0.9583333333x + x2 = 0.4379340278

Factor a perfect square on the left side:
(x + -0.4791666667)(x + -0.4791666667) = 0.4379340278

Calculate the square root of the right side: 0.661765841

Break this problem into two subproblems by setting 
(x + -0.4791666667) equal to 0.661765841 and -0.661765841.


Subproblem 1
x + -0.4791666667 = 0.661765841

Simplifying
x + -0.4791666667 = 0.661765841

Reorder the terms:
-0.4791666667 + x = 0.661765841

Solving
-0.4791666667 + x = 0.661765841

Solving for variable 'x'.

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

Add '0.4791666667' to each side of the equation.
-0.4791666667 + 0.4791666667 + x = 0.661765841 + 0.4791666667

Combine like terms: -0.4791666667 + 0.4791666667 = 0.0000000000
0.0000000000 + x = 0.661765841 + 0.4791666667
x = 0.661765841 + 0.4791666667

Combine like terms: 0.661765841 + 0.4791666667 = 1.1409325077
x = 1.1409325077

Simplifying
x = 1.1409325077


Subproblem 2
x + -0.4791666667 = -0.661765841

Simplifying
x + -0.4791666667 = -0.661765841

Reorder the terms:
-0.4791666667 + x = -0.661765841

Solving
-0.4791666667 + x = -0.661765841

Solving for variable 'x'.

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

Add '0.4791666667' to each side of the equation.
-0.4791666667 + 0.4791666667 + x = -0.661765841 + 0.4791666667

Combine like terms: -0.4791666667 + 0.4791666667 = 0.0000000000
0.0000000000 + x = -0.661765841 + 0.4791666667
x = -0.661765841 + 0.4791666667

Combine like terms: -0.661765841 + 0.4791666667 = -0.1825991743
x = -0.1825991743

Simplifying
x = -0.1825991743


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.1409325077, -0.1825991743}

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