r^2+5r=729

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Solution for r^2+5r=729 equation:


Simplifying
r2 + 5r = 729

Reorder the terms:
5r + r2 = 729

Solving
5r + r2 = 729

Solving for variable 'r'.

Reorder the terms:
-729 + 5r + r2 = 729 + -729

Combine like terms: 729 + -729 = 0
-729 + 5r + r2 = 0

Begin completing the square.

Move the constant term to the right:

Add '729' to each side of the equation.
-729 + 5r + 729 + r2 = 0 + 729

Reorder the terms:
-729 + 729 + 5r + r2 = 0 + 729

Combine like terms: -729 + 729 = 0
0 + 5r + r2 = 0 + 729
5r + r2 = 0 + 729

Combine like terms: 0 + 729 = 729
5r + r2 = 729

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 = 729 + 6.25

Reorder the terms:
6.25 + 5r + r2 = 729 + 6.25

Combine like terms: 729 + 6.25 = 735.25
6.25 + 5r + r2 = 735.25

Factor a perfect square on the left side:
(r + 2.5)(r + 2.5) = 735.25

Calculate the square root of the right side: 27.115493726

Break this problem into two subproblems by setting 
(r + 2.5) equal to 27.115493726 and -27.115493726.

Subproblem 1

r + 2.5 = 27.115493726 Simplifying r + 2.5 = 27.115493726 Reorder the terms: 2.5 + r = 27.115493726 Solving 2.5 + r = 27.115493726 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 = 27.115493726 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + r = 27.115493726 + -2.5 r = 27.115493726 + -2.5 Combine like terms: 27.115493726 + -2.5 = 24.615493726 r = 24.615493726 Simplifying r = 24.615493726

Subproblem 2

r + 2.5 = -27.115493726 Simplifying r + 2.5 = -27.115493726 Reorder the terms: 2.5 + r = -27.115493726 Solving 2.5 + r = -27.115493726 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 = -27.115493726 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + r = -27.115493726 + -2.5 r = -27.115493726 + -2.5 Combine like terms: -27.115493726 + -2.5 = -29.615493726 r = -29.615493726 Simplifying r = -29.615493726

Solution

The solution to the problem is based on the solutions from the subproblems. r = {24.615493726, -29.615493726}

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