(1+r)(1+r)=21

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Solution for (1+r)(1+r)=21 equation:


Simplifying
(1 + r)(1 + r) = 21

Multiply (1 + r) * (1 + r)
(1(1 + r) + r(1 + r)) = 21
((1 * 1 + r * 1) + r(1 + r)) = 21
((1 + 1r) + r(1 + r)) = 21
(1 + 1r + (1 * r + r * r)) = 21
(1 + 1r + (1r + r2)) = 21

Combine like terms: 1r + 1r = 2r
(1 + 2r + r2) = 21

Solving
1 + 2r + r2 = 21

Solving for variable 'r'.

Reorder the terms:
1 + -21 + 2r + r2 = 21 + -21

Combine like terms: 1 + -21 = -20
-20 + 2r + r2 = 21 + -21

Combine like terms: 21 + -21 = 0
-20 + 2r + r2 = 0

Begin completing the square.

Move the constant term to the right:

Add '20' to each side of the equation.
-20 + 2r + 20 + r2 = 0 + 20

Reorder the terms:
-20 + 20 + 2r + r2 = 0 + 20

Combine like terms: -20 + 20 = 0
0 + 2r + r2 = 0 + 20
2r + r2 = 0 + 20

Combine like terms: 0 + 20 = 20
2r + r2 = 20

The r term is 2r.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2r + 1 + r2 = 20 + 1

Reorder the terms:
1 + 2r + r2 = 20 + 1

Combine like terms: 20 + 1 = 21
1 + 2r + r2 = 21

Factor a perfect square on the left side:
(r + 1)(r + 1) = 21

Calculate the square root of the right side: 4.582575695

Break this problem into two subproblems by setting 
(r + 1) equal to 4.582575695 and -4.582575695.

Subproblem 1

r + 1 = 4.582575695 Simplifying r + 1 = 4.582575695 Reorder the terms: 1 + r = 4.582575695 Solving 1 + r = 4.582575695 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + r = 4.582575695 + -1 Combine like terms: 1 + -1 = 0 0 + r = 4.582575695 + -1 r = 4.582575695 + -1 Combine like terms: 4.582575695 + -1 = 3.582575695 r = 3.582575695 Simplifying r = 3.582575695

Subproblem 2

r + 1 = -4.582575695 Simplifying r + 1 = -4.582575695 Reorder the terms: 1 + r = -4.582575695 Solving 1 + r = -4.582575695 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + r = -4.582575695 + -1 Combine like terms: 1 + -1 = 0 0 + r = -4.582575695 + -1 r = -4.582575695 + -1 Combine like terms: -4.582575695 + -1 = -5.582575695 r = -5.582575695 Simplifying r = -5.582575695

Solution

The solution to the problem is based on the solutions from the subproblems. r = {3.582575695, -5.582575695}

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