24x^2-16x-89=0

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Solution for 24x^2-16x-89=0 equation: 


Simplifying
24x2 + -16x + -89 = 0

Reorder the terms:
-89 + -16x + 24x2 = 0

Solving
-89 + -16x + 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'.
-3.708333333 + -0.6666666667x + x2 = 0

Move the constant term to the right:

Add '3.708333333' to each side of the equation.
-3.708333333 + -0.6666666667x + 3.708333333 + x2 = 0 + 3.708333333

Reorder the terms:
-3.708333333 + 3.708333333 + -0.6666666667x + x2 = 0 + 3.708333333

Combine like terms: -3.708333333 + 3.708333333 = 0.000000000
0.000000000 + -0.6666666667x + x2 = 0 + 3.708333333
-0.6666666667x + x2 = 0 + 3.708333333

Combine like terms: 0 + 3.708333333 = 3.708333333
-0.6666666667x + x2 = 3.708333333

The x term is -0.6666666667x.  Take half its coefficient (-0.3333333334).
Square it (0.1111111112) and add it to both sides.

Add '0.1111111112' to each side of the equation.
-0.6666666667x + 0.1111111112 + x2 = 3.708333333 + 0.1111111112

Reorder the terms:
0.1111111112 + -0.6666666667x + x2 = 3.708333333 + 0.1111111112

Combine like terms: 3.708333333 + 0.1111111112 = 3.8194444442
0.1111111112 + -0.6666666667x + x2 = 3.8194444442

Factor a perfect square on the left side:
(x + -0.3333333334)(x + -0.3333333334) = 3.8194444442

Calculate the square root of the right side: 1.9543399

Break this problem into two subproblems by setting 
(x + -0.3333333334) equal to 1.9543399 and -1.9543399.


Subproblem 1
x + -0.3333333334 = 1.9543399

Simplifying
x + -0.3333333334 = 1.9543399

Reorder the terms:
-0.3333333334 + x = 1.9543399

Solving
-0.3333333334 + x = 1.9543399

Solving for variable 'x'.

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

Add '0.3333333334' to each side of the equation.
-0.3333333334 + 0.3333333334 + x = 1.9543399 + 0.3333333334

Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000
0.0000000000 + x = 1.9543399 + 0.3333333334
x = 1.9543399 + 0.3333333334

Combine like terms: 1.9543399 + 0.3333333334 = 2.2876732334
x = 2.2876732334

Simplifying
x = 2.2876732334


Subproblem 2
x + -0.3333333334 = -1.9543399

Simplifying
x + -0.3333333334 = -1.9543399

Reorder the terms:
-0.3333333334 + x = -1.9543399

Solving
-0.3333333334 + x = -1.9543399

Solving for variable 'x'.

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

Add '0.3333333334' to each side of the equation.
-0.3333333334 + 0.3333333334 + x = -1.9543399 + 0.3333333334

Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000
0.0000000000 + x = -1.9543399 + 0.3333333334
x = -1.9543399 + 0.3333333334

Combine like terms: -1.9543399 + 0.3333333334 = -1.6210065666
x = -1.6210065666

Simplifying
x = -1.6210065666


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.2876732334, -1.6210065666}

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