(x+5)(x+2)=31

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Solution for (x+5)(x+2)=31 equation:


Simplifying
(x + 5)(x + 2) = 31

Reorder the terms:
(5 + x)(x + 2) = 31

Reorder the terms:
(5 + x)(2 + x) = 31

Multiply (5 + x) * (2 + x)
(5(2 + x) + x(2 + x)) = 31
((2 * 5 + x * 5) + x(2 + x)) = 31
((10 + 5x) + x(2 + x)) = 31
(10 + 5x + (2 * x + x * x)) = 31
(10 + 5x + (2x + x2)) = 31

Combine like terms: 5x + 2x = 7x
(10 + 7x + x2) = 31

Solving
10 + 7x + x2 = 31

Solving for variable 'x'.

Reorder the terms:
10 + -31 + 7x + x2 = 31 + -31

Combine like terms: 10 + -31 = -21
-21 + 7x + x2 = 31 + -31

Combine like terms: 31 + -31 = 0
-21 + 7x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '21' to each side of the equation.
-21 + 7x + 21 + x2 = 0 + 21

Reorder the terms:
-21 + 21 + 7x + x2 = 0 + 21

Combine like terms: -21 + 21 = 0
0 + 7x + x2 = 0 + 21
7x + x2 = 0 + 21

Combine like terms: 0 + 21 = 21
7x + x2 = 21

The x term is 7x.  Take half its coefficient (3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
7x + 12.25 + x2 = 21 + 12.25

Reorder the terms:
12.25 + 7x + x2 = 21 + 12.25

Combine like terms: 21 + 12.25 = 33.25
12.25 + 7x + x2 = 33.25

Factor a perfect square on the left side:
(x + 3.5)(x + 3.5) = 33.25

Calculate the square root of the right side: 5.766281297

Break this problem into two subproblems by setting 
(x + 3.5) equal to 5.766281297 and -5.766281297.

Subproblem 1

x + 3.5 = 5.766281297 Simplifying x + 3.5 = 5.766281297 Reorder the terms: 3.5 + x = 5.766281297 Solving 3.5 + x = 5.766281297 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + x = 5.766281297 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + x = 5.766281297 + -3.5 x = 5.766281297 + -3.5 Combine like terms: 5.766281297 + -3.5 = 2.266281297 x = 2.266281297 Simplifying x = 2.266281297

Subproblem 2

x + 3.5 = -5.766281297 Simplifying x + 3.5 = -5.766281297 Reorder the terms: 3.5 + x = -5.766281297 Solving 3.5 + x = -5.766281297 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + x = -5.766281297 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + x = -5.766281297 + -3.5 x = -5.766281297 + -3.5 Combine like terms: -5.766281297 + -3.5 = -9.266281297 x = -9.266281297 Simplifying x = -9.266281297

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.266281297, -9.266281297}

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