(2x+3)(x+5)=5

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


Simplifying
(2x + 3)(x + 5) = 5

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

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

Multiply (3 + 2x) * (5 + x)
(3(5 + x) + 2x * (5 + x)) = 5
((5 * 3 + x * 3) + 2x * (5 + x)) = 5
((15 + 3x) + 2x * (5 + x)) = 5
(15 + 3x + (5 * 2x + x * 2x)) = 5
(15 + 3x + (10x + 2x2)) = 5

Combine like terms: 3x + 10x = 13x
(15 + 13x + 2x2) = 5

Solving
15 + 13x + 2x2 = 5

Solving for variable 'x'.

Reorder the terms:
15 + -5 + 13x + 2x2 = 5 + -5

Combine like terms: 15 + -5 = 10
10 + 13x + 2x2 = 5 + -5

Combine like terms: 5 + -5 = 0
10 + 13x + 2x2 = 0

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

Divide each side by '2'.
5 + 6.5x + x2 = 0

Move the constant term to the right:

Add '-5' to each side of the equation.
5 + 6.5x + -5 + x2 = 0 + -5

Reorder the terms:
5 + -5 + 6.5x + x2 = 0 + -5

Combine like terms: 5 + -5 = 0
0 + 6.5x + x2 = 0 + -5
6.5x + x2 = 0 + -5

Combine like terms: 0 + -5 = -5
6.5x + x2 = -5

The x term is 6.5x.  Take half its coefficient (3.25).
Square it (10.5625) and add it to both sides.

Add '10.5625' to each side of the equation.
6.5x + 10.5625 + x2 = -5 + 10.5625

Reorder the terms:
10.5625 + 6.5x + x2 = -5 + 10.5625

Combine like terms: -5 + 10.5625 = 5.5625
10.5625 + 6.5x + x2 = 5.5625

Factor a perfect square on the left side:
(x + 3.25)(x + 3.25) = 5.5625

Calculate the square root of the right side: 2.358495283

Break this problem into two subproblems by setting 
(x + 3.25) equal to 2.358495283 and -2.358495283.

Subproblem 1

x + 3.25 = 2.358495283 Simplifying x + 3.25 = 2.358495283 Reorder the terms: 3.25 + x = 2.358495283 Solving 3.25 + x = 2.358495283 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.25' to each side of the equation. 3.25 + -3.25 + x = 2.358495283 + -3.25 Combine like terms: 3.25 + -3.25 = 0.00 0.00 + x = 2.358495283 + -3.25 x = 2.358495283 + -3.25 Combine like terms: 2.358495283 + -3.25 = -0.891504717 x = -0.891504717 Simplifying x = -0.891504717

Subproblem 2

x + 3.25 = -2.358495283 Simplifying x + 3.25 = -2.358495283 Reorder the terms: 3.25 + x = -2.358495283 Solving 3.25 + x = -2.358495283 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.25' to each side of the equation. 3.25 + -3.25 + x = -2.358495283 + -3.25 Combine like terms: 3.25 + -3.25 = 0.00 0.00 + x = -2.358495283 + -3.25 x = -2.358495283 + -3.25 Combine like terms: -2.358495283 + -3.25 = -5.608495283 x = -5.608495283 Simplifying x = -5.608495283

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.891504717, -5.608495283}

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