12r+2r^2=45

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Solution for 12r+2r^2=45 equation:


Simplifying
12r + 2r2 = 45

Solving
12r + 2r2 = 45

Solving for variable 'r'.

Reorder the terms:
-45 + 12r + 2r2 = 45 + -45

Combine like terms: 45 + -45 = 0
-45 + 12r + 2r2 = 0

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

Divide each side by '2'.
-22.5 + 6r + r2 = 0

Move the constant term to the right:

Add '22.5' to each side of the equation.
-22.5 + 6r + 22.5 + r2 = 0 + 22.5

Reorder the terms:
-22.5 + 22.5 + 6r + r2 = 0 + 22.5

Combine like terms: -22.5 + 22.5 = 0.0
0.0 + 6r + r2 = 0 + 22.5
6r + r2 = 0 + 22.5

Combine like terms: 0 + 22.5 = 22.5
6r + r2 = 22.5

The r term is 6r.  Take half its coefficient (3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
6r + 9 + r2 = 22.5 + 9

Reorder the terms:
9 + 6r + r2 = 22.5 + 9

Combine like terms: 22.5 + 9 = 31.5
9 + 6r + r2 = 31.5

Factor a perfect square on the left side:
(r + 3)(r + 3) = 31.5

Calculate the square root of the right side: 5.61248608

Break this problem into two subproblems by setting 
(r + 3) equal to 5.61248608 and -5.61248608.

Subproblem 1

r + 3 = 5.61248608 Simplifying r + 3 = 5.61248608 Reorder the terms: 3 + r = 5.61248608 Solving 3 + r = 5.61248608 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + r = 5.61248608 + -3 Combine like terms: 3 + -3 = 0 0 + r = 5.61248608 + -3 r = 5.61248608 + -3 Combine like terms: 5.61248608 + -3 = 2.61248608 r = 2.61248608 Simplifying r = 2.61248608

Subproblem 2

r + 3 = -5.61248608 Simplifying r + 3 = -5.61248608 Reorder the terms: 3 + r = -5.61248608 Solving 3 + r = -5.61248608 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + r = -5.61248608 + -3 Combine like terms: 3 + -3 = 0 0 + r = -5.61248608 + -3 r = -5.61248608 + -3 Combine like terms: -5.61248608 + -3 = -8.61248608 r = -8.61248608 Simplifying r = -8.61248608

Solution

The solution to the problem is based on the solutions from the subproblems. r = {2.61248608, -8.61248608}

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