6r^2+19r=10

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Solution for 6r^2+19r=10 equation:


Simplifying
6r2 + 19r = 10

Reorder the terms:
19r + 6r2 = 10

Solving
19r + 6r2 = 10

Solving for variable 'r'.

Reorder the terms:
-10 + 19r + 6r2 = 10 + -10

Combine like terms: 10 + -10 = 0
-10 + 19r + 6r2 = 0

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

Divide each side by '6'.
-1.666666667 + 3.166666667r + r2 = 0

Move the constant term to the right:

Add '1.666666667' to each side of the equation.
-1.666666667 + 3.166666667r + 1.666666667 + r2 = 0 + 1.666666667

Reorder the terms:
-1.666666667 + 1.666666667 + 3.166666667r + r2 = 0 + 1.666666667

Combine like terms: -1.666666667 + 1.666666667 = 0.000000000
0.000000000 + 3.166666667r + r2 = 0 + 1.666666667
3.166666667r + r2 = 0 + 1.666666667

Combine like terms: 0 + 1.666666667 = 1.666666667
3.166666667r + r2 = 1.666666667

The r term is 3.166666667r.  Take half its coefficient (1.583333334).
Square it (2.506944447) and add it to both sides.

Add '2.506944447' to each side of the equation.
3.166666667r + 2.506944447 + r2 = 1.666666667 + 2.506944447

Reorder the terms:
2.506944447 + 3.166666667r + r2 = 1.666666667 + 2.506944447

Combine like terms: 1.666666667 + 2.506944447 = 4.173611114
2.506944447 + 3.166666667r + r2 = 4.173611114

Factor a perfect square on the left side:
(r + 1.583333334)(r + 1.583333334) = 4.173611114

Calculate the square root of the right side: 2.042941779

Break this problem into two subproblems by setting 
(r + 1.583333334) equal to 2.042941779 and -2.042941779.

Subproblem 1

r + 1.583333334 = 2.042941779 Simplifying r + 1.583333334 = 2.042941779 Reorder the terms: 1.583333334 + r = 2.042941779 Solving 1.583333334 + r = 2.042941779 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1.583333334' to each side of the equation. 1.583333334 + -1.583333334 + r = 2.042941779 + -1.583333334 Combine like terms: 1.583333334 + -1.583333334 = 0.000000000 0.000000000 + r = 2.042941779 + -1.583333334 r = 2.042941779 + -1.583333334 Combine like terms: 2.042941779 + -1.583333334 = 0.459608445 r = 0.459608445 Simplifying r = 0.459608445

Subproblem 2

r + 1.583333334 = -2.042941779 Simplifying r + 1.583333334 = -2.042941779 Reorder the terms: 1.583333334 + r = -2.042941779 Solving 1.583333334 + r = -2.042941779 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1.583333334' to each side of the equation. 1.583333334 + -1.583333334 + r = -2.042941779 + -1.583333334 Combine like terms: 1.583333334 + -1.583333334 = 0.000000000 0.000000000 + r = -2.042941779 + -1.583333334 r = -2.042941779 + -1.583333334 Combine like terms: -2.042941779 + -1.583333334 = -3.626275113 r = -3.626275113 Simplifying r = -3.626275113

Solution

The solution to the problem is based on the solutions from the subproblems. r = {0.459608445, -3.626275113}

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