r(r+5)=23

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Solution for r(r+5)=23 equation:


Simplifying
r(r + 5) = 23

Reorder the terms:
r(5 + r) = 23
(5 * r + r * r) = 23
(5r + r2) = 23

Solving
5r + r2 = 23

Solving for variable 'r'.

Reorder the terms:
-23 + 5r + r2 = 23 + -23

Combine like terms: 23 + -23 = 0
-23 + 5r + r2 = 0

Begin completing the square.

Move the constant term to the right:

Add '23' to each side of the equation.
-23 + 5r + 23 + r2 = 0 + 23

Reorder the terms:
-23 + 23 + 5r + r2 = 0 + 23

Combine like terms: -23 + 23 = 0
0 + 5r + r2 = 0 + 23
5r + r2 = 0 + 23

Combine like terms: 0 + 23 = 23
5r + r2 = 23

The r term is 5r.  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.
5r + 6.25 + r2 = 23 + 6.25

Reorder the terms:
6.25 + 5r + r2 = 23 + 6.25

Combine like terms: 23 + 6.25 = 29.25
6.25 + 5r + r2 = 29.25

Factor a perfect square on the left side:
(r + 2.5)(r + 2.5) = 29.25

Calculate the square root of the right side: 5.408326913

Break this problem into two subproblems by setting 
(r + 2.5) equal to 5.408326913 and -5.408326913.

Subproblem 1

r + 2.5 = 5.408326913 Simplifying r + 2.5 = 5.408326913 Reorder the terms: 2.5 + r = 5.408326913 Solving 2.5 + r = 5.408326913 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + r = 5.408326913 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + r = 5.408326913 + -2.5 r = 5.408326913 + -2.5 Combine like terms: 5.408326913 + -2.5 = 2.908326913 r = 2.908326913 Simplifying r = 2.908326913

Subproblem 2

r + 2.5 = -5.408326913 Simplifying r + 2.5 = -5.408326913 Reorder the terms: 2.5 + r = -5.408326913 Solving 2.5 + r = -5.408326913 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + r = -5.408326913 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + r = -5.408326913 + -2.5 r = -5.408326913 + -2.5 Combine like terms: -5.408326913 + -2.5 = -7.908326913 r = -7.908326913 Simplifying r = -7.908326913

Solution

The solution to the problem is based on the solutions from the subproblems. r = {2.908326913, -7.908326913}

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