15x(5+x)=40

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Solution for 15x(5+x)=40 equation: 


Simplifying
15x(5 + x) = 40
(5 * 15x + x * 15x) = 40
(75x + 15x2) = 40

Solving
75x + 15x2 = 40

Solving for variable 'x'.

Reorder the terms:
-40 + 75x + 15x2 = 40 + -40

Combine like terms: 40 + -40 = 0
-40 + 75x + 15x2 = 0

Factor out the Greatest Common Factor (GCF), '5'.
5(-8 + 15x + 3x2) = 0

Ignore the factor 5.

Subproblem 1
Set the factor '(-8 + 15x + 3x2)' equal to zero and attempt to solve:

Simplifying
-8 + 15x + 3x2 = 0

Solving
-8 + 15x + 3x2 = 0

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

Divide each side by '3'.
-2.666666667 + 5x + x2 = 0

Move the constant term to the right:

Add '2.666666667' to each side of the equation.
-2.666666667 + 5x + 2.666666667 + x2 = 0 + 2.666666667

Reorder the terms:
-2.666666667 + 2.666666667 + 5x + x2 = 0 + 2.666666667

Combine like terms: -2.666666667 + 2.666666667 = 0.000000000
0.000000000 + 5x + x2 = 0 + 2.666666667
5x + x2 = 0 + 2.666666667

Combine like terms: 0 + 2.666666667 = 2.666666667
5x + x2 = 2.666666667

The x term is 5x.  Take half its coefficient (2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
5x + 6.25 + x2 = 2.666666667 + 6.25

Reorder the terms:
6.25 + 5x + x2 = 2.666666667 + 6.25

Combine like terms: 2.666666667 + 6.25 = 8.916666667
6.25 + 5x + x2 = 8.916666667

Factor a perfect square on the left side:
(x + 2.5)(x + 2.5) = 8.916666667

Calculate the square root of the right side: 2.986078811

Break this problem into two subproblems by setting 
(x + 2.5) equal to 2.986078811 and -2.986078811.


Subproblem 1
x + 2.5 = 2.986078811

Simplifying
x + 2.5 = 2.986078811

Reorder the terms:
2.5 + x = 2.986078811

Solving
2.5 + x = 2.986078811

Solving for variable 'x'.

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

Add '-2.5' to each side of the equation.
2.5 + -2.5 + x = 2.986078811 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + x = 2.986078811 + -2.5
x = 2.986078811 + -2.5

Combine like terms: 2.986078811 + -2.5 = 0.486078811
x = 0.486078811

Simplifying
x = 0.486078811


Subproblem 2
x + 2.5 = -2.986078811

Simplifying
x + 2.5 = -2.986078811

Reorder the terms:
2.5 + x = -2.986078811

Solving
2.5 + x = -2.986078811

Solving for variable 'x'.

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

Add '-2.5' to each side of the equation.
2.5 + -2.5 + x = -2.986078811 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + x = -2.986078811 + -2.5
x = -2.986078811 + -2.5

Combine like terms: -2.986078811 + -2.5 = -5.486078811
x = -5.486078811

Simplifying
x = -5.486078811


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.486078811, -5.486078811}
Solution
x = {0.486078811, -5.486078811}

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