10x(x+4)=5(x+13)

Solution for 10x(x+4)=5(x+13) equation:

Simplifying
10x(x + 4) = 5(x + 13)

Reorder the terms:
10x(4 + x) = 5(x + 13)
(4 * 10x + x * 10x) = 5(x + 13)
(40x + 10x2) = 5(x + 13)

Reorder the terms:
40x + 10x2 = 5(13 + x)
40x + 10x2 = (13 * 5 + x * 5)
40x + 10x2 = (65 + 5x)

Solving
40x + 10x2 = 65 + 5x

Solving for variable 'x'.

Reorder the terms:
-65 + 40x + -5x + 10x2 = 65 + 5x + -65 + -5x

Combine like terms: 40x + -5x = 35x
-65 + 35x + 10x2 = 65 + 5x + -65 + -5x

Reorder the terms:
-65 + 35x + 10x2 = 65 + -65 + 5x + -5x

Combine like terms: 65 + -65 = 0
-65 + 35x + 10x2 = 0 + 5x + -5x
-65 + 35x + 10x2 = 5x + -5x

Combine like terms: 5x + -5x = 0
-65 + 35x + 10x2 = 0

Factor out the Greatest Common Factor (GCF), '5'.
5(-13 + 7x + 2x2) = 0

Ignore the factor 5.
Subproblem 1
Set the factor '(-13 + 7x + 2x2)' equal to zero and attempt to solve:

Simplifying
-13 + 7x + 2x2 = 0

Solving
-13 + 7x + 2x2 = 0

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

Divide each side by '2'.
-6.5 + 3.5x + x2 = 0

Move the constant term to the right:

Add '6.5' to each side of the equation.
-6.5 + 3.5x + 6.5 + x2 = 0 + 6.5

Reorder the terms:
-6.5 + 6.5 + 3.5x + x2 = 0 + 6.5

Combine like terms: -6.5 + 6.5 = 0.0
0.0 + 3.5x + x2 = 0 + 6.5
3.5x + x2 = 0 + 6.5

Combine like terms: 0 + 6.5 = 6.5
3.5x + x2 = 6.5

The x term is 3.5x.  Take half its coefficient (1.75).
Square it (3.0625) and add it to both sides.

Add '3.0625' to each side of the equation.
3.5x + 3.0625 + x2 = 6.5 + 3.0625

Reorder the terms:
3.0625 + 3.5x + x2 = 6.5 + 3.0625

Combine like terms: 6.5 + 3.0625 = 9.5625
3.0625 + 3.5x + x2 = 9.5625

Factor a perfect square on the left side:
(x + 1.75)(x + 1.75) = 9.5625

Calculate the square root of the right side: 3.092329219

Break this problem into two subproblems by setting 
(x + 1.75) equal to 3.092329219 and -3.092329219.

Subproblem 1
x + 1.75 = 3.092329219

Simplifying
x + 1.75 = 3.092329219

Reorder the terms:
1.75 + x = 3.092329219

Solving
1.75 + x = 3.092329219

Solving for variable 'x'.

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

Add '-1.75' to each side of the equation.
1.75 + -1.75 + x = 3.092329219 + -1.75

Combine like terms: 1.75 + -1.75 = 0.00
0.00 + x = 3.092329219 + -1.75
x = 3.092329219 + -1.75

Combine like terms: 3.092329219 + -1.75 = 1.342329219
x = 1.342329219

Simplifying
x = 1.342329219

Subproblem 2
x + 1.75 = -3.092329219

Simplifying
x + 1.75 = -3.092329219

Reorder the terms:
1.75 + x = -3.092329219

Solving
1.75 + x = -3.092329219

Solving for variable 'x'.

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

Add '-1.75' to each side of the equation.
1.75 + -1.75 + x = -3.092329219 + -1.75

Combine like terms: 1.75 + -1.75 = 0.00
0.00 + x = -3.092329219 + -1.75
x = -3.092329219 + -1.75

Combine like terms: -3.092329219 + -1.75 = -4.842329219
x = -4.842329219

Simplifying
x = -4.842329219

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.342329219, -4.842329219}
Solution
x = {1.342329219, -4.842329219}