(r-2)(r+8)=23

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Solution for (r-2)(r+8)=23 equation:


Simplifying
(r + -2)(r + 8) = 23

Reorder the terms:
(-2 + r)(r + 8) = 23

Reorder the terms:
(-2 + r)(8 + r) = 23

Multiply (-2 + r) * (8 + r)
(-2(8 + r) + r(8 + r)) = 23
((8 * -2 + r * -2) + r(8 + r)) = 23
((-16 + -2r) + r(8 + r)) = 23
(-16 + -2r + (8 * r + r * r)) = 23
(-16 + -2r + (8r + r2)) = 23

Combine like terms: -2r + 8r = 6r
(-16 + 6r + r2) = 23

Solving
-16 + 6r + r2 = 23

Solving for variable 'r'.

Reorder the terms:
-16 + -23 + 6r + r2 = 23 + -23

Combine like terms: -16 + -23 = -39
-39 + 6r + r2 = 23 + -23

Combine like terms: 23 + -23 = 0
-39 + 6r + r2 = 0

Begin completing the square.

Move the constant term to the right:

Add '39' to each side of the equation.
-39 + 6r + 39 + r2 = 0 + 39

Reorder the terms:
-39 + 39 + 6r + r2 = 0 + 39

Combine like terms: -39 + 39 = 0
0 + 6r + r2 = 0 + 39
6r + r2 = 0 + 39

Combine like terms: 0 + 39 = 39
6r + r2 = 39

The r term is 6r.  Take half its coefficient (3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
6r + 9 + r2 = 39 + 9

Reorder the terms:
9 + 6r + r2 = 39 + 9

Combine like terms: 39 + 9 = 48
9 + 6r + r2 = 48

Factor a perfect square on the left side:
(r + 3)(r + 3) = 48

Calculate the square root of the right side: 6.92820323

Break this problem into two subproblems by setting 
(r + 3) equal to 6.92820323 and -6.92820323.

Subproblem 1

r + 3 = 6.92820323 Simplifying r + 3 = 6.92820323 Reorder the terms: 3 + r = 6.92820323 Solving 3 + r = 6.92820323 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + r = 6.92820323 + -3 Combine like terms: 3 + -3 = 0 0 + r = 6.92820323 + -3 r = 6.92820323 + -3 Combine like terms: 6.92820323 + -3 = 3.92820323 r = 3.92820323 Simplifying r = 3.92820323

Subproblem 2

r + 3 = -6.92820323 Simplifying r + 3 = -6.92820323 Reorder the terms: 3 + r = -6.92820323 Solving 3 + r = -6.92820323 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + r = -6.92820323 + -3 Combine like terms: 3 + -3 = 0 0 + r = -6.92820323 + -3 r = -6.92820323 + -3 Combine like terms: -6.92820323 + -3 = -9.92820323 r = -9.92820323 Simplifying r = -9.92820323

Solution

The solution to the problem is based on the solutions from the subproblems. r = {3.92820323, -9.92820323}

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