21r^2-133r-20=0

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Solution for 21r^2-133r-20=0 equation:


Simplifying
21r2 + -133r + -20 = 0

Reorder the terms:
-20 + -133r + 21r2 = 0

Solving
-20 + -133r + 21r2 = 0

Solving for variable 'r'.

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

Divide each side by '21'.
-0.9523809524 + -6.333333333r + r2 = 0

Move the constant term to the right:

Add '0.9523809524' to each side of the equation.
-0.9523809524 + -6.333333333r + 0.9523809524 + r2 = 0 + 0.9523809524

Reorder the terms:
-0.9523809524 + 0.9523809524 + -6.333333333r + r2 = 0 + 0.9523809524

Combine like terms: -0.9523809524 + 0.9523809524 = 0.0000000000
0.0000000000 + -6.333333333r + r2 = 0 + 0.9523809524
-6.333333333r + r2 = 0 + 0.9523809524

Combine like terms: 0 + 0.9523809524 = 0.9523809524
-6.333333333r + r2 = 0.9523809524

The r term is -6.333333333r.  Take half its coefficient (-3.166666667).
Square it (10.02777778) and add it to both sides.

Add '10.02777778' to each side of the equation.
-6.333333333r + 10.02777778 + r2 = 0.9523809524 + 10.02777778

Reorder the terms:
10.02777778 + -6.333333333r + r2 = 0.9523809524 + 10.02777778

Combine like terms: 0.9523809524 + 10.02777778 = 10.9801587324
10.02777778 + -6.333333333r + r2 = 10.9801587324

Factor a perfect square on the left side:
(r + -3.166666667)(r + -3.166666667) = 10.9801587324

Calculate the square root of the right side: 3.313632257

Break this problem into two subproblems by setting 
(r + -3.166666667) equal to 3.313632257 and -3.313632257.

Subproblem 1

r + -3.166666667 = 3.313632257 Simplifying r + -3.166666667 = 3.313632257 Reorder the terms: -3.166666667 + r = 3.313632257 Solving -3.166666667 + r = 3.313632257 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '3.166666667' to each side of the equation. -3.166666667 + 3.166666667 + r = 3.313632257 + 3.166666667 Combine like terms: -3.166666667 + 3.166666667 = 0.000000000 0.000000000 + r = 3.313632257 + 3.166666667 r = 3.313632257 + 3.166666667 Combine like terms: 3.313632257 + 3.166666667 = 6.480298924 r = 6.480298924 Simplifying r = 6.480298924

Subproblem 2

r + -3.166666667 = -3.313632257 Simplifying r + -3.166666667 = -3.313632257 Reorder the terms: -3.166666667 + r = -3.313632257 Solving -3.166666667 + r = -3.313632257 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '3.166666667' to each side of the equation. -3.166666667 + 3.166666667 + r = -3.313632257 + 3.166666667 Combine like terms: -3.166666667 + 3.166666667 = 0.000000000 0.000000000 + r = -3.313632257 + 3.166666667 r = -3.313632257 + 3.166666667 Combine like terms: -3.313632257 + 3.166666667 = -0.14696559 r = -0.14696559 Simplifying r = -0.14696559

Solution

The solution to the problem is based on the solutions from the subproblems. r = {6.480298924, -0.14696559}

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