6r(7r-3)+3r(10r+9)=1

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Solution for 6r(7r-3)+3r(10r+9)=1 equation:


Simplifying
6r(7r + -3) + 3r(10r + 9) = 1

Reorder the terms:
6r(-3 + 7r) + 3r(10r + 9) = 1
(-3 * 6r + 7r * 6r) + 3r(10r + 9) = 1
(-18r + 42r2) + 3r(10r + 9) = 1

Reorder the terms:
-18r + 42r2 + 3r(9 + 10r) = 1
-18r + 42r2 + (9 * 3r + 10r * 3r) = 1
-18r + 42r2 + (27r + 30r2) = 1

Reorder the terms:
-18r + 27r + 42r2 + 30r2 = 1

Combine like terms: -18r + 27r = 9r
9r + 42r2 + 30r2 = 1

Combine like terms: 42r2 + 30r2 = 72r2
9r + 72r2 = 1

Solving
9r + 72r2 = 1

Solving for variable 'r'.

Reorder the terms:
-1 + 9r + 72r2 = 1 + -1

Combine like terms: 1 + -1 = 0
-1 + 9r + 72r2 = 0

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

Divide each side by '72'.
-0.01388888889 + 0.125r + r2 = 0

Move the constant term to the right:

Add '0.01388888889' to each side of the equation.
-0.01388888889 + 0.125r + 0.01388888889 + r2 = 0 + 0.01388888889

Reorder the terms:
-0.01388888889 + 0.01388888889 + 0.125r + r2 = 0 + 0.01388888889

Combine like terms: -0.01388888889 + 0.01388888889 = 0.00000000000
0.00000000000 + 0.125r + r2 = 0 + 0.01388888889
0.125r + r2 = 0 + 0.01388888889

Combine like terms: 0 + 0.01388888889 = 0.01388888889
0.125r + r2 = 0.01388888889

The r term is 0.125r.  Take half its coefficient (0.0625).
Square it (0.00390625) and add it to both sides.

Add '0.00390625' to each side of the equation.
0.125r + 0.00390625 + r2 = 0.01388888889 + 0.00390625

Reorder the terms:
0.00390625 + 0.125r + r2 = 0.01388888889 + 0.00390625

Combine like terms: 0.01388888889 + 0.00390625 = 0.01779513889
0.00390625 + 0.125r + r2 = 0.01779513889

Factor a perfect square on the left side:
(r + 0.0625)(r + 0.0625) = 0.01779513889

Calculate the square root of the right side: 0.133398422

Break this problem into two subproblems by setting 
(r + 0.0625) equal to 0.133398422 and -0.133398422.

Subproblem 1

r + 0.0625 = 0.133398422 Simplifying r + 0.0625 = 0.133398422 Reorder the terms: 0.0625 + r = 0.133398422 Solving 0.0625 + r = 0.133398422 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.0625' to each side of the equation. 0.0625 + -0.0625 + r = 0.133398422 + -0.0625 Combine like terms: 0.0625 + -0.0625 = 0.0000 0.0000 + r = 0.133398422 + -0.0625 r = 0.133398422 + -0.0625 Combine like terms: 0.133398422 + -0.0625 = 0.070898422 r = 0.070898422 Simplifying r = 0.070898422

Subproblem 2

r + 0.0625 = -0.133398422 Simplifying r + 0.0625 = -0.133398422 Reorder the terms: 0.0625 + r = -0.133398422 Solving 0.0625 + r = -0.133398422 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.0625' to each side of the equation. 0.0625 + -0.0625 + r = -0.133398422 + -0.0625 Combine like terms: 0.0625 + -0.0625 = 0.0000 0.0000 + r = -0.133398422 + -0.0625 r = -0.133398422 + -0.0625 Combine like terms: -0.133398422 + -0.0625 = -0.195898422 r = -0.195898422 Simplifying r = -0.195898422

Solution

The solution to the problem is based on the solutions from the subproblems. r = {0.070898422, -0.195898422}

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