4x(x+5)+10(x+1)=24

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Solution for 4x(x+5)+10(x+1)=24 equation: 


Simplifying
4x(x + 5) + 10(x + 1) = 24

Reorder the terms:
4x(5 + x) + 10(x + 1) = 24
(5 * 4x + x * 4x) + 10(x + 1) = 24
(20x + 4x2) + 10(x + 1) = 24

Reorder the terms:
20x + 4x2 + 10(1 + x) = 24
20x + 4x2 + (1 * 10 + x * 10) = 24
20x + 4x2 + (10 + 10x) = 24

Reorder the terms:
10 + 20x + 10x + 4x2 = 24

Combine like terms: 20x + 10x = 30x
10 + 30x + 4x2 = 24

Solving
10 + 30x + 4x2 = 24

Solving for variable 'x'.

Reorder the terms:
10 + -24 + 30x + 4x2 = 24 + -24

Combine like terms: 10 + -24 = -14
-14 + 30x + 4x2 = 24 + -24

Combine like terms: 24 + -24 = 0
-14 + 30x + 4x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-7 + 15x + 2x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-7 + 15x + 2x2)' equal to zero and attempt to solve:

Simplifying
-7 + 15x + 2x2 = 0

Solving
-7 + 15x + 2x2 = 0

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

Divide each side by '2'.
-3.5 + 7.5x + x2 = 0

Move the constant term to the right:

Add '3.5' to each side of the equation.
-3.5 + 7.5x + 3.5 + x2 = 0 + 3.5

Reorder the terms:
-3.5 + 3.5 + 7.5x + x2 = 0 + 3.5

Combine like terms: -3.5 + 3.5 = 0.0
0.0 + 7.5x + x2 = 0 + 3.5
7.5x + x2 = 0 + 3.5

Combine like terms: 0 + 3.5 = 3.5
7.5x + x2 = 3.5

The x term is 7.5x.  Take half its coefficient (3.75).
Square it (14.0625) and add it to both sides.

Add '14.0625' to each side of the equation.
7.5x + 14.0625 + x2 = 3.5 + 14.0625

Reorder the terms:
14.0625 + 7.5x + x2 = 3.5 + 14.0625

Combine like terms: 3.5 + 14.0625 = 17.5625
14.0625 + 7.5x + x2 = 17.5625

Factor a perfect square on the left side:
(x + 3.75)(x + 3.75) = 17.5625

Calculate the square root of the right side: 4.190763654

Break this problem into two subproblems by setting 
(x + 3.75) equal to 4.190763654 and -4.190763654.


Subproblem 1
x + 3.75 = 4.190763654

Simplifying
x + 3.75 = 4.190763654

Reorder the terms:
3.75 + x = 4.190763654

Solving
3.75 + x = 4.190763654

Solving for variable 'x'.

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

Add '-3.75' to each side of the equation.
3.75 + -3.75 + x = 4.190763654 + -3.75

Combine like terms: 3.75 + -3.75 = 0.00
0.00 + x = 4.190763654 + -3.75
x = 4.190763654 + -3.75

Combine like terms: 4.190763654 + -3.75 = 0.440763654
x = 0.440763654

Simplifying
x = 0.440763654


Subproblem 2
x + 3.75 = -4.190763654

Simplifying
x + 3.75 = -4.190763654

Reorder the terms:
3.75 + x = -4.190763654

Solving
3.75 + x = -4.190763654

Solving for variable 'x'.

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

Add '-3.75' to each side of the equation.
3.75 + -3.75 + x = -4.190763654 + -3.75

Combine like terms: 3.75 + -3.75 = 0.00
0.00 + x = -4.190763654 + -3.75
x = -4.190763654 + -3.75

Combine like terms: -4.190763654 + -3.75 = -7.940763654
x = -7.940763654

Simplifying
x = -7.940763654


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.440763654, -7.940763654}
Solution
x = {0.440763654, -7.940763654}

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