(x+2)(x+3)=84

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Solution for (x+2)(x+3)=84 equation:


Simplifying
(x + 2)(x + 3) = 84

Reorder the terms:
(2 + x)(x + 3) = 84

Reorder the terms:
(2 + x)(3 + x) = 84

Multiply (2 + x) * (3 + x)
(2(3 + x) + x(3 + x)) = 84
((3 * 2 + x * 2) + x(3 + x)) = 84
((6 + 2x) + x(3 + x)) = 84
(6 + 2x + (3 * x + x * x)) = 84
(6 + 2x + (3x + x2)) = 84

Combine like terms: 2x + 3x = 5x
(6 + 5x + x2) = 84

Solving
6 + 5x + x2 = 84

Solving for variable 'x'.

Reorder the terms:
6 + -84 + 5x + x2 = 84 + -84

Combine like terms: 6 + -84 = -78
-78 + 5x + x2 = 84 + -84

Combine like terms: 84 + -84 = 0
-78 + 5x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '78' to each side of the equation.
-78 + 5x + 78 + x2 = 0 + 78

Reorder the terms:
-78 + 78 + 5x + x2 = 0 + 78

Combine like terms: -78 + 78 = 0
0 + 5x + x2 = 0 + 78
5x + x2 = 0 + 78

Combine like terms: 0 + 78 = 78
5x + x2 = 78

The x term is 5x.  Take half its coefficient (2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
5x + 6.25 + x2 = 78 + 6.25

Reorder the terms:
6.25 + 5x + x2 = 78 + 6.25

Combine like terms: 78 + 6.25 = 84.25
6.25 + 5x + x2 = 84.25

Factor a perfect square on the left side:
(x + 2.5)(x + 2.5) = 84.25

Calculate the square root of the right side: 9.178779875

Break this problem into two subproblems by setting 
(x + 2.5) equal to 9.178779875 and -9.178779875.

Subproblem 1

x + 2.5 = 9.178779875 Simplifying x + 2.5 = 9.178779875 Reorder the terms: 2.5 + x = 9.178779875 Solving 2.5 + x = 9.178779875 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + x = 9.178779875 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + x = 9.178779875 + -2.5 x = 9.178779875 + -2.5 Combine like terms: 9.178779875 + -2.5 = 6.678779875 x = 6.678779875 Simplifying x = 6.678779875

Subproblem 2

x + 2.5 = -9.178779875 Simplifying x + 2.5 = -9.178779875 Reorder the terms: 2.5 + x = -9.178779875 Solving 2.5 + x = -9.178779875 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + x = -9.178779875 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + x = -9.178779875 + -2.5 x = -9.178779875 + -2.5 Combine like terms: -9.178779875 + -2.5 = -11.678779875 x = -11.678779875 Simplifying x = -11.678779875

Solution

The solution to the problem is based on the solutions from the subproblems. x = {6.678779875, -11.678779875}

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