3r^2-7r=8

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Solution for 3r^2-7r=8 equation: 


Simplifying
3r2 + -7r = 8

Reorder the terms:
-7r + 3r2 = 8

Solving
-7r + 3r2 = 8

Solving for variable 'r'.

Reorder the terms:
-8 + -7r + 3r2 = 8 + -8

Combine like terms: 8 + -8 = 0
-8 + -7r + 3r2 = 0

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

Divide each side by '3'.
-2.666666667 + -2.333333333r + r2 = 0

Move the constant term to the right:

Add '2.666666667' to each side of the equation.
-2.666666667 + -2.333333333r + 2.666666667 + r2 = 0 + 2.666666667

Reorder the terms:
-2.666666667 + 2.666666667 + -2.333333333r + r2 = 0 + 2.666666667

Combine like terms: -2.666666667 + 2.666666667 = 0.000000000
0.000000000 + -2.333333333r + r2 = 0 + 2.666666667
-2.333333333r + r2 = 0 + 2.666666667

Combine like terms: 0 + 2.666666667 = 2.666666667
-2.333333333r + r2 = 2.666666667

The r term is -2.333333333r.  Take half its coefficient (-1.166666667).
Square it (1.361111112) and add it to both sides.

Add '1.361111112' to each side of the equation.
-2.333333333r + 1.361111112 + r2 = 2.666666667 + 1.361111112

Reorder the terms:
1.361111112 + -2.333333333r + r2 = 2.666666667 + 1.361111112

Combine like terms: 2.666666667 + 1.361111112 = 4.027777779
1.361111112 + -2.333333333r + r2 = 4.027777779

Factor a perfect square on the left side:
(r + -1.166666667)(r + -1.166666667) = 4.027777779

Calculate the square root of the right side: 2.00693243

Break this problem into two subproblems by setting 
(r + -1.166666667) equal to 2.00693243 and -2.00693243.


Subproblem 1
r + -1.166666667 = 2.00693243

Simplifying
r + -1.166666667 = 2.00693243

Reorder the terms:
-1.166666667 + r = 2.00693243

Solving
-1.166666667 + r = 2.00693243

Solving for variable 'r'.

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

Add '1.166666667' to each side of the equation.
-1.166666667 + 1.166666667 + r = 2.00693243 + 1.166666667

Combine like terms: -1.166666667 + 1.166666667 = 0.000000000
0.000000000 + r = 2.00693243 + 1.166666667
r = 2.00693243 + 1.166666667

Combine like terms: 2.00693243 + 1.166666667 = 3.173599097
r = 3.173599097

Simplifying
r = 3.173599097


Subproblem 2
r + -1.166666667 = -2.00693243

Simplifying
r + -1.166666667 = -2.00693243

Reorder the terms:
-1.166666667 + r = -2.00693243

Solving
-1.166666667 + r = -2.00693243

Solving for variable 'r'.

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

Add '1.166666667' to each side of the equation.
-1.166666667 + 1.166666667 + r = -2.00693243 + 1.166666667

Combine like terms: -1.166666667 + 1.166666667 = 0.000000000
0.000000000 + r = -2.00693243 + 1.166666667
r = -2.00693243 + 1.166666667

Combine like terms: -2.00693243 + 1.166666667 = -0.840265763
r = -0.840265763

Simplifying
r = -0.840265763


Solution
The solution to the problem is based on the solutions
from the subproblems.
r = {3.173599097, -0.840265763}

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