29r=3(r^2+4)

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Solution for 29r=3(r^2+4) equation:


Simplifying
29r = 3(r2 + 4)

Reorder the terms:
29r = 3(4 + r2)
29r = (4 * 3 + r2 * 3)
29r = (12 + 3r2)

Solving
29r = 12 + 3r2

Solving for variable 'r'.

Reorder the terms:
-12 + 29r + -3r2 = 12 + 3r2 + -12 + -3r2

Reorder the terms:
-12 + 29r + -3r2 = 12 + -12 + 3r2 + -3r2

Combine like terms: 12 + -12 = 0
-12 + 29r + -3r2 = 0 + 3r2 + -3r2
-12 + 29r + -3r2 = 3r2 + -3r2

Combine like terms: 3r2 + -3r2 = 0
-12 + 29r + -3r2 = 0

Begin completing the square.  Divide all terms by
-3 the coefficient of the squared term: 

Divide each side by '-3'.
4 + -9.666666667r + r2 = 0

Move the constant term to the right:

Add '-4' to each side of the equation.
4 + -9.666666667r + -4 + r2 = 0 + -4

Reorder the terms:
4 + -4 + -9.666666667r + r2 = 0 + -4

Combine like terms: 4 + -4 = 0
0 + -9.666666667r + r2 = 0 + -4
-9.666666667r + r2 = 0 + -4

Combine like terms: 0 + -4 = -4
-9.666666667r + r2 = -4

The r term is -9.666666667r.  Take half its coefficient (-4.833333334).
Square it (23.36111112) and add it to both sides.

Add '23.36111112' to each side of the equation.
-9.666666667r + 23.36111112 + r2 = -4 + 23.36111112

Reorder the terms:
23.36111112 + -9.666666667r + r2 = -4 + 23.36111112

Combine like terms: -4 + 23.36111112 = 19.36111112
23.36111112 + -9.666666667r + r2 = 19.36111112

Factor a perfect square on the left side:
(r + -4.833333334)(r + -4.833333334) = 19.36111112

Calculate the square root of the right side: 4.400126262

Break this problem into two subproblems by setting 
(r + -4.833333334) equal to 4.400126262 and -4.400126262.

Subproblem 1

r + -4.833333334 = 4.400126262 Simplifying r + -4.833333334 = 4.400126262 Reorder the terms: -4.833333334 + r = 4.400126262 Solving -4.833333334 + r = 4.400126262 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '4.833333334' to each side of the equation. -4.833333334 + 4.833333334 + r = 4.400126262 + 4.833333334 Combine like terms: -4.833333334 + 4.833333334 = 0.000000000 0.000000000 + r = 4.400126262 + 4.833333334 r = 4.400126262 + 4.833333334 Combine like terms: 4.400126262 + 4.833333334 = 9.233459596 r = 9.233459596 Simplifying r = 9.233459596

Subproblem 2

r + -4.833333334 = -4.400126262 Simplifying r + -4.833333334 = -4.400126262 Reorder the terms: -4.833333334 + r = -4.400126262 Solving -4.833333334 + r = -4.400126262 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '4.833333334' to each side of the equation. -4.833333334 + 4.833333334 + r = -4.400126262 + 4.833333334 Combine like terms: -4.833333334 + 4.833333334 = 0.000000000 0.000000000 + r = -4.400126262 + 4.833333334 r = -4.400126262 + 4.833333334 Combine like terms: -4.400126262 + 4.833333334 = 0.433207072 r = 0.433207072 Simplifying r = 0.433207072

Solution

The solution to the problem is based on the solutions from the subproblems. r = {9.233459596, 0.433207072}

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