21r^2+3r-2=0

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Solution for 21r^2+3r-2=0 equation:


Simplifying
21r2 + 3r + -2 = 0

Reorder the terms:
-2 + 3r + 21r2 = 0

Solving
-2 + 3r + 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'.
-0.09523809524 + 0.1428571429r + r2 = 0

Move the constant term to the right:

Add '0.09523809524' to each side of the equation.
-0.09523809524 + 0.1428571429r + 0.09523809524 + r2 = 0 + 0.09523809524

Reorder the terms:
-0.09523809524 + 0.09523809524 + 0.1428571429r + r2 = 0 + 0.09523809524

Combine like terms: -0.09523809524 + 0.09523809524 = 0.00000000000
0.00000000000 + 0.1428571429r + r2 = 0 + 0.09523809524
0.1428571429r + r2 = 0 + 0.09523809524

Combine like terms: 0 + 0.09523809524 = 0.09523809524
0.1428571429r + r2 = 0.09523809524

The r term is 0.1428571429r.  Take half its coefficient (0.07142857145).
Square it (0.005102040819) and add it to both sides.

Add '0.005102040819' to each side of the equation.
0.1428571429r + 0.005102040819 + r2 = 0.09523809524 + 0.005102040819

Reorder the terms:
0.005102040819 + 0.1428571429r + r2 = 0.09523809524 + 0.005102040819

Combine like terms: 0.09523809524 + 0.005102040819 = 0.100340136059
0.005102040819 + 0.1428571429r + r2 = 0.100340136059

Factor a perfect square on the left side:
(r + 0.07142857145)(r + 0.07142857145) = 0.100340136059

Calculate the square root of the right side: 0.316765112

Break this problem into two subproblems by setting 
(r + 0.07142857145) equal to 0.316765112 and -0.316765112.

Subproblem 1

r + 0.07142857145 = 0.316765112 Simplifying r + 0.07142857145 = 0.316765112 Reorder the terms: 0.07142857145 + r = 0.316765112 Solving 0.07142857145 + r = 0.316765112 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.07142857145' to each side of the equation. 0.07142857145 + -0.07142857145 + r = 0.316765112 + -0.07142857145 Combine like terms: 0.07142857145 + -0.07142857145 = 0.00000000000 0.00000000000 + r = 0.316765112 + -0.07142857145 r = 0.316765112 + -0.07142857145 Combine like terms: 0.316765112 + -0.07142857145 = 0.24533654055 r = 0.24533654055 Simplifying r = 0.24533654055

Subproblem 2

r + 0.07142857145 = -0.316765112 Simplifying r + 0.07142857145 = -0.316765112 Reorder the terms: 0.07142857145 + r = -0.316765112 Solving 0.07142857145 + r = -0.316765112 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.07142857145' to each side of the equation. 0.07142857145 + -0.07142857145 + r = -0.316765112 + -0.07142857145 Combine like terms: 0.07142857145 + -0.07142857145 = 0.00000000000 0.00000000000 + r = -0.316765112 + -0.07142857145 r = -0.316765112 + -0.07142857145 Combine like terms: -0.316765112 + -0.07142857145 = -0.38819368345 r = -0.38819368345 Simplifying r = -0.38819368345

Solution

The solution to the problem is based on the solutions from the subproblems. r = {0.24533654055, -0.38819368345}

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