8=r^2+r

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Solution for 8=r^2+r equation:


Simplifying
8 = r2 + r

Reorder the terms:
8 = r + r2

Solving
8 = r + r2

Solving for variable 'r'.

Reorder the terms:
8 + -1r + -1r2 = r + -1r + r2 + -1r2

Combine like terms: r + -1r = 0
8 + -1r + -1r2 = 0 + r2 + -1r2
8 + -1r + -1r2 = r2 + -1r2

Combine like terms: r2 + -1r2 = 0
8 + -1r + -1r2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-8 + r + r2 = 0

Move the constant term to the right:

Add '8' to each side of the equation.
-8 + r + 8 + r2 = 0 + 8

Reorder the terms:
-8 + 8 + r + r2 = 0 + 8

Combine like terms: -8 + 8 = 0
0 + r + r2 = 0 + 8
r + r2 = 0 + 8

Combine like terms: 0 + 8 = 8
r + r2 = 8

The r term is r.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
 + 0.25 + r2 = 8 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + r2 = 8 + 0.25

Combine like terms: 8 + 0.25 = 8.25
1.25 + r2 = 8.25

Factor a perfect square on the left side:
(r + 0.5)(r + 0.5) = 8.25

Calculate the square root of the right side: 2.872281323

Break this problem into two subproblems by setting 
(r + 0.5) equal to 2.872281323 and -2.872281323.

Subproblem 1

r + 0.5 = 2.872281323 Simplifying r + 0.5 = 2.872281323 Reorder the terms: 0.5 + r = 2.872281323 Solving 0.5 + r = 2.872281323 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + r = 2.872281323 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + r = 2.872281323 + -0.5 r = 2.872281323 + -0.5 Combine like terms: 2.872281323 + -0.5 = 2.372281323 r = 2.372281323 Simplifying r = 2.372281323

Subproblem 2

r + 0.5 = -2.872281323 Simplifying r + 0.5 = -2.872281323 Reorder the terms: 0.5 + r = -2.872281323 Solving 0.5 + r = -2.872281323 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + r = -2.872281323 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + r = -2.872281323 + -0.5 r = -2.872281323 + -0.5 Combine like terms: -2.872281323 + -0.5 = -3.372281323 r = -3.372281323 Simplifying r = -3.372281323

Solution

The solution to the problem is based on the solutions from the subproblems. r = {2.372281323, -3.372281323}

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