3r+9r(12-2r)=2

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Solution for 3r+9r(12-2r)=2 equation:


Simplifying
3r + 9r(12 + -2r) = 2
3r + (12 * 9r + -2r * 9r) = 2
3r + (108r + -18r2) = 2

Combine like terms: 3r + 108r = 111r
111r + -18r2 = 2

Solving
111r + -18r2 = 2

Solving for variable 'r'.

Reorder the terms:
-2 + 111r + -18r2 = 2 + -2

Combine like terms: 2 + -2 = 0
-2 + 111r + -18r2 = 0

Begin completing the square.  Divide all terms by
-18 the coefficient of the squared term: 

Divide each side by '-18'.
0.1111111111 + -6.166666667r + r2 = 0

Move the constant term to the right:

Add '-0.1111111111' to each side of the equation.
0.1111111111 + -6.166666667r + -0.1111111111 + r2 = 0 + -0.1111111111

Reorder the terms:
0.1111111111 + -0.1111111111 + -6.166666667r + r2 = 0 + -0.1111111111

Combine like terms: 0.1111111111 + -0.1111111111 = 0.0000000000
0.0000000000 + -6.166666667r + r2 = 0 + -0.1111111111
-6.166666667r + r2 = 0 + -0.1111111111

Combine like terms: 0 + -0.1111111111 = -0.1111111111
-6.166666667r + r2 = -0.1111111111

The r term is -6.166666667r.  Take half its coefficient (-3.083333334).
Square it (9.506944449) and add it to both sides.

Add '9.506944449' to each side of the equation.
-6.166666667r + 9.506944449 + r2 = -0.1111111111 + 9.506944449

Reorder the terms:
9.506944449 + -6.166666667r + r2 = -0.1111111111 + 9.506944449

Combine like terms: -0.1111111111 + 9.506944449 = 9.3958333379
9.506944449 + -6.166666667r + r2 = 9.3958333379

Factor a perfect square on the left side:
(r + -3.083333334)(r + -3.083333334) = 9.3958333379

Calculate the square root of the right side: 3.06526236

Break this problem into two subproblems by setting 
(r + -3.083333334) equal to 3.06526236 and -3.06526236.

Subproblem 1

r + -3.083333334 = 3.06526236 Simplifying r + -3.083333334 = 3.06526236 Reorder the terms: -3.083333334 + r = 3.06526236 Solving -3.083333334 + r = 3.06526236 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '3.083333334' to each side of the equation. -3.083333334 + 3.083333334 + r = 3.06526236 + 3.083333334 Combine like terms: -3.083333334 + 3.083333334 = 0.000000000 0.000000000 + r = 3.06526236 + 3.083333334 r = 3.06526236 + 3.083333334 Combine like terms: 3.06526236 + 3.083333334 = 6.148595694 r = 6.148595694 Simplifying r = 6.148595694

Subproblem 2

r + -3.083333334 = -3.06526236 Simplifying r + -3.083333334 = -3.06526236 Reorder the terms: -3.083333334 + r = -3.06526236 Solving -3.083333334 + r = -3.06526236 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '3.083333334' to each side of the equation. -3.083333334 + 3.083333334 + r = -3.06526236 + 3.083333334 Combine like terms: -3.083333334 + 3.083333334 = 0.000000000 0.000000000 + r = -3.06526236 + 3.083333334 r = -3.06526236 + 3.083333334 Combine like terms: -3.06526236 + 3.083333334 = 0.018070974 r = 0.018070974 Simplifying r = 0.018070974

Solution

The solution to the problem is based on the solutions from the subproblems. r = {6.148595694, 0.018070974}

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