(x+3)(x+5)+(5-x)(2x+1)=0

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Solution for (x+3)(x+5)+(5-x)(2x+1)=0 equation:


Simplifying
(x + 3)(x + 5) + (5 + -1x)(2x + 1) = 0

Reorder the terms:
(3 + x)(x + 5) + (5 + -1x)(2x + 1) = 0

Reorder the terms:
(3 + x)(5 + x) + (5 + -1x)(2x + 1) = 0

Multiply (3 + x) * (5 + x)
(3(5 + x) + x(5 + x)) + (5 + -1x)(2x + 1) = 0
((5 * 3 + x * 3) + x(5 + x)) + (5 + -1x)(2x + 1) = 0
((15 + 3x) + x(5 + x)) + (5 + -1x)(2x + 1) = 0
(15 + 3x + (5 * x + x * x)) + (5 + -1x)(2x + 1) = 0
(15 + 3x + (5x + x2)) + (5 + -1x)(2x + 1) = 0

Combine like terms: 3x + 5x = 8x
(15 + 8x + x2) + (5 + -1x)(2x + 1) = 0

Reorder the terms:
15 + 8x + x2 + (5 + -1x)(1 + 2x) = 0

Multiply (5 + -1x) * (1 + 2x)
15 + 8x + x2 + (5(1 + 2x) + -1x * (1 + 2x)) = 0
15 + 8x + x2 + ((1 * 5 + 2x * 5) + -1x * (1 + 2x)) = 0
15 + 8x + x2 + ((5 + 10x) + -1x * (1 + 2x)) = 0
15 + 8x + x2 + (5 + 10x + (1 * -1x + 2x * -1x)) = 0
15 + 8x + x2 + (5 + 10x + (-1x + -2x2)) = 0

Combine like terms: 10x + -1x = 9x
15 + 8x + x2 + (5 + 9x + -2x2) = 0

Reorder the terms:
15 + 5 + 8x + 9x + x2 + -2x2 = 0

Combine like terms: 15 + 5 = 20
20 + 8x + 9x + x2 + -2x2 = 0

Combine like terms: 8x + 9x = 17x
20 + 17x + x2 + -2x2 = 0

Combine like terms: x2 + -2x2 = -1x2
20 + 17x + -1x2 = 0

Solving
20 + 17x + -1x2 = 0

Solving for variable 'x'.

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-20 + -17x + x2 = 0

Move the constant term to the right:

Add '20' to each side of the equation.
-20 + -17x + 20 + x2 = 0 + 20

Reorder the terms:
-20 + 20 + -17x + x2 = 0 + 20

Combine like terms: -20 + 20 = 0
0 + -17x + x2 = 0 + 20
-17x + x2 = 0 + 20

Combine like terms: 0 + 20 = 20
-17x + x2 = 20

The x term is -17x.  Take half its coefficient (-8.5).
Square it (72.25) and add it to both sides.

Add '72.25' to each side of the equation.
-17x + 72.25 + x2 = 20 + 72.25

Reorder the terms:
72.25 + -17x + x2 = 20 + 72.25

Combine like terms: 20 + 72.25 = 92.25
72.25 + -17x + x2 = 92.25

Factor a perfect square on the left side:
(x + -8.5)(x + -8.5) = 92.25

Calculate the square root of the right side: 9.604686356

Break this problem into two subproblems by setting 
(x + -8.5) equal to 9.604686356 and -9.604686356.

Subproblem 1

x + -8.5 = 9.604686356 Simplifying x + -8.5 = 9.604686356 Reorder the terms: -8.5 + x = 9.604686356 Solving -8.5 + x = 9.604686356 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '8.5' to each side of the equation. -8.5 + 8.5 + x = 9.604686356 + 8.5 Combine like terms: -8.5 + 8.5 = 0.0 0.0 + x = 9.604686356 + 8.5 x = 9.604686356 + 8.5 Combine like terms: 9.604686356 + 8.5 = 18.104686356 x = 18.104686356 Simplifying x = 18.104686356

Subproblem 2

x + -8.5 = -9.604686356 Simplifying x + -8.5 = -9.604686356 Reorder the terms: -8.5 + x = -9.604686356 Solving -8.5 + x = -9.604686356 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '8.5' to each side of the equation. -8.5 + 8.5 + x = -9.604686356 + 8.5 Combine like terms: -8.5 + 8.5 = 0.0 0.0 + x = -9.604686356 + 8.5 x = -9.604686356 + 8.5 Combine like terms: -9.604686356 + 8.5 = -1.104686356 x = -1.104686356 Simplifying x = -1.104686356

Solution

The solution to the problem is based on the solutions from the subproblems. x = {18.104686356, -1.104686356}

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