3r^2+17r-84=0

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Solution for 3r^2+17r-84=0 equation: 


Simplifying
3r2 + 17r + -84 = 0

Reorder the terms:
-84 + 17r + 3r2 = 0

Solving
-84 + 17r + 3r2 = 0

Solving for variable 'r'.

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

Divide each side by '3'.
-28 + 5.666666667r + r2 = 0

Move the constant term to the right:

Add '28' to each side of the equation.
-28 + 5.666666667r + 28 + r2 = 0 + 28

Reorder the terms:
-28 + 28 + 5.666666667r + r2 = 0 + 28

Combine like terms: -28 + 28 = 0
0 + 5.666666667r + r2 = 0 + 28
5.666666667r + r2 = 0 + 28

Combine like terms: 0 + 28 = 28
5.666666667r + r2 = 28

The r term is 5.666666667r.  Take half its coefficient (2.833333334).
Square it (8.027777782) and add it to both sides.

Add '8.027777782' to each side of the equation.
5.666666667r + 8.027777782 + r2 = 28 + 8.027777782

Reorder the terms:
8.027777782 + 5.666666667r + r2 = 28 + 8.027777782

Combine like terms: 28 + 8.027777782 = 36.027777782
8.027777782 + 5.666666667r + r2 = 36.027777782

Factor a perfect square on the left side:
(r + 2.833333334)(r + 2.833333334) = 36.027777782

Calculate the square root of the right side: 6.002314369

Break this problem into two subproblems by setting 
(r + 2.833333334) equal to 6.002314369 and -6.002314369.


Subproblem 1
r + 2.833333334 = 6.002314369

Simplifying
r + 2.833333334 = 6.002314369

Reorder the terms:
2.833333334 + r = 6.002314369

Solving
2.833333334 + r = 6.002314369

Solving for variable 'r'.

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

Add '-2.833333334' to each side of the equation.
2.833333334 + -2.833333334 + r = 6.002314369 + -2.833333334

Combine like terms: 2.833333334 + -2.833333334 = 0.000000000
0.000000000 + r = 6.002314369 + -2.833333334
r = 6.002314369 + -2.833333334

Combine like terms: 6.002314369 + -2.833333334 = 3.168981035
r = 3.168981035

Simplifying
r = 3.168981035


Subproblem 2
r + 2.833333334 = -6.002314369

Simplifying
r + 2.833333334 = -6.002314369

Reorder the terms:
2.833333334 + r = -6.002314369

Solving
2.833333334 + r = -6.002314369

Solving for variable 'r'.

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

Add '-2.833333334' to each side of the equation.
2.833333334 + -2.833333334 + r = -6.002314369 + -2.833333334

Combine like terms: 2.833333334 + -2.833333334 = 0.000000000
0.000000000 + r = -6.002314369 + -2.833333334
r = -6.002314369 + -2.833333334

Combine like terms: -6.002314369 + -2.833333334 = -8.835647703
r = -8.835647703

Simplifying
r = -8.835647703


Solution
The solution to the problem is based on the solutions
from the subproblems.
r = {3.168981035, -8.835647703}

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