x(3x+5)=24

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Solution for x(3x+5)=24 equation: 


Simplifying
x(3x + 5) = 24

Reorder the terms:
x(5 + 3x) = 24
(5 * x + 3x * x) = 24
(5x + 3x2) = 24

Solving
5x + 3x2 = 24

Solving for variable 'x'.

Reorder the terms:
-24 + 5x + 3x2 = 24 + -24

Combine like terms: 24 + -24 = 0
-24 + 5x + 3x2 = 0

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

Divide each side by '3'.
-8 + 1.666666667x + x2 = 0

Move the constant term to the right:

Add '8' to each side of the equation.
-8 + 1.666666667x + 8 + x2 = 0 + 8

Reorder the terms:
-8 + 8 + 1.666666667x + x2 = 0 + 8

Combine like terms: -8 + 8 = 0
0 + 1.666666667x + x2 = 0 + 8
1.666666667x + x2 = 0 + 8

Combine like terms: 0 + 8 = 8
1.666666667x + x2 = 8

The x term is 1.666666667x.  Take half its coefficient (0.8333333335).
Square it (0.6944444447) and add it to both sides.

Add '0.6944444447' to each side of the equation.
1.666666667x + 0.6944444447 + x2 = 8 + 0.6944444447

Reorder the terms:
0.6944444447 + 1.666666667x + x2 = 8 + 0.6944444447

Combine like terms: 8 + 0.6944444447 = 8.6944444447
0.6944444447 + 1.666666667x + x2 = 8.6944444447

Factor a perfect square on the left side:
(x + 0.8333333335)(x + 0.8333333335) = 8.6944444447

Calculate the square root of the right side: 2.948634336

Break this problem into two subproblems by setting 
(x + 0.8333333335) equal to 2.948634336 and -2.948634336.


Subproblem 1
x + 0.8333333335 = 2.948634336

Simplifying
x + 0.8333333335 = 2.948634336

Reorder the terms:
0.8333333335 + x = 2.948634336

Solving
0.8333333335 + x = 2.948634336

Solving for variable 'x'.

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

Add '-0.8333333335' to each side of the equation.
0.8333333335 + -0.8333333335 + x = 2.948634336 + -0.8333333335

Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000
0.0000000000 + x = 2.948634336 + -0.8333333335
x = 2.948634336 + -0.8333333335

Combine like terms: 2.948634336 + -0.8333333335 = 2.1153010025
x = 2.1153010025

Simplifying
x = 2.1153010025


Subproblem 2
x + 0.8333333335 = -2.948634336

Simplifying
x + 0.8333333335 = -2.948634336

Reorder the terms:
0.8333333335 + x = -2.948634336

Solving
0.8333333335 + x = -2.948634336

Solving for variable 'x'.

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

Add '-0.8333333335' to each side of the equation.
0.8333333335 + -0.8333333335 + x = -2.948634336 + -0.8333333335

Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000
0.0000000000 + x = -2.948634336 + -0.8333333335
x = -2.948634336 + -0.8333333335

Combine like terms: -2.948634336 + -0.8333333335 = -3.7819676695
x = -3.7819676695

Simplifying
x = -3.7819676695


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.1153010025, -3.7819676695}

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