r(r+5)=51

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Solution for r(r+5)=51 equation:


Simplifying
r(r + 5) = 51

Reorder the terms:
r(5 + r) = 51
(5 * r + r * r) = 51
(5r + r2) = 51

Solving
5r + r2 = 51

Solving for variable 'r'.

Reorder the terms:
-51 + 5r + r2 = 51 + -51

Combine like terms: 51 + -51 = 0
-51 + 5r + r2 = 0

Begin completing the square.

Move the constant term to the right:

Add '51' to each side of the equation.
-51 + 5r + 51 + r2 = 0 + 51

Reorder the terms:
-51 + 51 + 5r + r2 = 0 + 51

Combine like terms: -51 + 51 = 0
0 + 5r + r2 = 0 + 51
5r + r2 = 0 + 51

Combine like terms: 0 + 51 = 51
5r + r2 = 51

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 = 51 + 6.25

Reorder the terms:
6.25 + 5r + r2 = 51 + 6.25

Combine like terms: 51 + 6.25 = 57.25
6.25 + 5r + r2 = 57.25

Factor a perfect square on the left side:
(r + 2.5)(r + 2.5) = 57.25

Calculate the square root of the right side: 7.566372975

Break this problem into two subproblems by setting 
(r + 2.5) equal to 7.566372975 and -7.566372975.


Subproblem 1
r + 2.5 = 7.566372975

Simplifying
r + 2.5 = 7.566372975

Reorder the terms:
2.5 + r = 7.566372975

Solving
2.5 + r = 7.566372975

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 = 7.566372975 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + r = 7.566372975 + -2.5
r = 7.566372975 + -2.5

Combine like terms: 7.566372975 + -2.5 = 5.066372975
r = 5.066372975

Simplifying
r = 5.066372975


Subproblem 2
r + 2.5 = -7.566372975

Simplifying
r + 2.5 = -7.566372975

Reorder the terms:
2.5 + r = -7.566372975

Solving
2.5 + r = -7.566372975

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 = -7.566372975 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + r = -7.566372975 + -2.5
r = -7.566372975 + -2.5

Combine like terms: -7.566372975 + -2.5 = -10.066372975
r = -10.066372975

Simplifying
r = -10.066372975


Solution
The solution to the problem is based on the solutions
from the subproblems.
r = {5.066372975, -10.066372975}

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