3r^2+8r=21

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Solution for 3r^2+8r=21 equation:


Simplifying
3r2 + 8r = 21

Reorder the terms:
8r + 3r2 = 21

Solving
8r + 3r2 = 21

Solving for variable 'r'.

Reorder the terms:
-21 + 8r + 3r2 = 21 + -21

Combine like terms: 21 + -21 = 0
-21 + 8r + 3r2 = 0

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

Divide each side by '3'.
-7 + 2.666666667r + r2 = 0

Move the constant term to the right:

Add '7' to each side of the equation.
-7 + 2.666666667r + 7 + r2 = 0 + 7

Reorder the terms:
-7 + 7 + 2.666666667r + r2 = 0 + 7

Combine like terms: -7 + 7 = 0
0 + 2.666666667r + r2 = 0 + 7
2.666666667r + r2 = 0 + 7

Combine like terms: 0 + 7 = 7
2.666666667r + r2 = 7

The r term is 2.666666667r.  Take half its coefficient (1.333333334).
Square it (1.777777780) and add it to both sides.

Add '1.777777780' to each side of the equation.
2.666666667r + 1.777777780 + r2 = 7 + 1.777777780

Reorder the terms:
1.777777780 + 2.666666667r + r2 = 7 + 1.777777780

Combine like terms: 7 + 1.777777780 = 8.77777778
1.777777780 + 2.666666667r + r2 = 8.77777778

Factor a perfect square on the left side:
(r + 1.333333334)(r + 1.333333334) = 8.77777778

Calculate the square root of the right side: 2.962731473

Break this problem into two subproblems by setting 
(r + 1.333333334) equal to 2.962731473 and -2.962731473.

Subproblem 1

r + 1.333333334 = 2.962731473 Simplifying r + 1.333333334 = 2.962731473 Reorder the terms: 1.333333334 + r = 2.962731473 Solving 1.333333334 + r = 2.962731473 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1.333333334' to each side of the equation. 1.333333334 + -1.333333334 + r = 2.962731473 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + r = 2.962731473 + -1.333333334 r = 2.962731473 + -1.333333334 Combine like terms: 2.962731473 + -1.333333334 = 1.629398139 r = 1.629398139 Simplifying r = 1.629398139

Subproblem 2

r + 1.333333334 = -2.962731473 Simplifying r + 1.333333334 = -2.962731473 Reorder the terms: 1.333333334 + r = -2.962731473 Solving 1.333333334 + r = -2.962731473 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1.333333334' to each side of the equation. 1.333333334 + -1.333333334 + r = -2.962731473 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + r = -2.962731473 + -1.333333334 r = -2.962731473 + -1.333333334 Combine like terms: -2.962731473 + -1.333333334 = -4.296064807 r = -4.296064807 Simplifying r = -4.296064807

Solution

The solution to the problem is based on the solutions from the subproblems. r = {1.629398139, -4.296064807}

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