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

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


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

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

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

Multiply (-2 + x) * (3 + x)
(-2(3 + x) + x(3 + x)) = (x + 5) * 2
((3 * -2 + x * -2) + x(3 + x)) = (x + 5) * 2
((-6 + -2x) + x(3 + x)) = (x + 5) * 2
(-6 + -2x + (3 * x + x * x)) = (x + 5) * 2
(-6 + -2x + (3x + x2)) = (x + 5) * 2

Combine like terms: -2x + 3x = 1x
(-6 + 1x + x2) = (x + 5) * 2

Reorder the terms:
-6 + 1x + x2 = (5 + x) * 2

Reorder the terms for easier multiplication:
-6 + 1x + x2 = 2(5 + x)
-6 + 1x + x2 = (5 * 2 + x * 2)
-6 + 1x + x2 = (10 + 2x)

Solving
-6 + 1x + x2 = 10 + 2x

Solving for variable 'x'.

Reorder the terms:
-6 + -10 + 1x + -2x + x2 = 10 + 2x + -10 + -2x

Combine like terms: -6 + -10 = -16
-16 + 1x + -2x + x2 = 10 + 2x + -10 + -2x

Combine like terms: 1x + -2x = -1x
-16 + -1x + x2 = 10 + 2x + -10 + -2x

Reorder the terms:
-16 + -1x + x2 = 10 + -10 + 2x + -2x

Combine like terms: 10 + -10 = 0
-16 + -1x + x2 = 0 + 2x + -2x
-16 + -1x + x2 = 2x + -2x

Combine like terms: 2x + -2x = 0
-16 + -1x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '16' to each side of the equation.
-16 + -1x + 16 + x2 = 0 + 16

Reorder the terms:
-16 + 16 + -1x + x2 = 0 + 16

Combine like terms: -16 + 16 = 0
0 + -1x + x2 = 0 + 16
-1x + x2 = 0 + 16

Combine like terms: 0 + 16 = 16
-1x + x2 = 16

The x term is -1x.  Take half its coefficient (-0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1x + 0.25 + x2 = 16 + 0.25

Reorder the terms:
0.25 + -1x + x2 = 16 + 0.25

Combine like terms: 16 + 0.25 = 16.25
0.25 + -1x + x2 = 16.25

Factor a perfect square on the left side:
(x + -0.5)(x + -0.5) = 16.25

Calculate the square root of the right side: 4.031128874

Break this problem into two subproblems by setting 
(x + -0.5) equal to 4.031128874 and -4.031128874.

Subproblem 1

x + -0.5 = 4.031128874 Simplifying x + -0.5 = 4.031128874 Reorder the terms: -0.5 + x = 4.031128874 Solving -0.5 + x = 4.031128874 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x = 4.031128874 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = 4.031128874 + 0.5 x = 4.031128874 + 0.5 Combine like terms: 4.031128874 + 0.5 = 4.531128874 x = 4.531128874 Simplifying x = 4.531128874

Subproblem 2

x + -0.5 = -4.031128874 Simplifying x + -0.5 = -4.031128874 Reorder the terms: -0.5 + x = -4.031128874 Solving -0.5 + x = -4.031128874 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x = -4.031128874 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = -4.031128874 + 0.5 x = -4.031128874 + 0.5 Combine like terms: -4.031128874 + 0.5 = -3.531128874 x = -3.531128874 Simplifying x = -3.531128874

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.531128874, -3.531128874}

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