r^2+21r-90=0

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Solution for r^2+21r-90=0 equation: 


Simplifying
r2 + 21r + -90 = 0

Reorder the terms:
-90 + 21r + r2 = 0

Solving
-90 + 21r + r2 = 0

Solving for variable 'r'.

Begin completing the square.

Move the constant term to the right:

Add '90' to each side of the equation.
-90 + 21r + 90 + r2 = 0 + 90

Reorder the terms:
-90 + 90 + 21r + r2 = 0 + 90

Combine like terms: -90 + 90 = 0
0 + 21r + r2 = 0 + 90
21r + r2 = 0 + 90

Combine like terms: 0 + 90 = 90
21r + r2 = 90

The r term is 21r.  Take half its coefficient (10.5).
Square it (110.25) and add it to both sides.

Add '110.25' to each side of the equation.
21r + 110.25 + r2 = 90 + 110.25

Reorder the terms:
110.25 + 21r + r2 = 90 + 110.25

Combine like terms: 90 + 110.25 = 200.25
110.25 + 21r + r2 = 200.25

Factor a perfect square on the left side:
(r + 10.5)(r + 10.5) = 200.25

Calculate the square root of the right side: 14.150971698

Break this problem into two subproblems by setting 
(r + 10.5) equal to 14.150971698 and -14.150971698.


Subproblem 1
r + 10.5 = 14.150971698

Simplifying
r + 10.5 = 14.150971698

Reorder the terms:
10.5 + r = 14.150971698

Solving
10.5 + r = 14.150971698

Solving for variable 'r'.

Move all terms containing r to the left, all other terms to the right.

Add '-10.5' to each side of the equation.
10.5 + -10.5 + r = 14.150971698 + -10.5

Combine like terms: 10.5 + -10.5 = 0.0
0.0 + r = 14.150971698 + -10.5
r = 14.150971698 + -10.5

Combine like terms: 14.150971698 + -10.5 = 3.650971698
r = 3.650971698

Simplifying
r = 3.650971698


Subproblem 2
r + 10.5 = -14.150971698

Simplifying
r + 10.5 = -14.150971698

Reorder the terms:
10.5 + r = -14.150971698

Solving
10.5 + r = -14.150971698

Solving for variable 'r'.

Move all terms containing r to the left, all other terms to the right.

Add '-10.5' to each side of the equation.
10.5 + -10.5 + r = -14.150971698 + -10.5

Combine like terms: 10.5 + -10.5 = 0.0
0.0 + r = -14.150971698 + -10.5
r = -14.150971698 + -10.5

Combine like terms: -14.150971698 + -10.5 = -24.650971698
r = -24.650971698

Simplifying
r = -24.650971698


Solution
The solution to the problem is based on the solutions
from the subproblems.
r = {3.650971698, -24.650971698}

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