3r^2+46r-120=0

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Solution for 3r^2+46r-120=0 equation:


Simplifying
3r2 + 46r + -120 = 0

Reorder the terms:
-120 + 46r + 3r2 = 0

Solving
-120 + 46r + 3r2 = 0

Solving for variable 'r'.

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

Divide each side by '3'.
-40 + 15.33333333r + r2 = 0

Move the constant term to the right:

Add '40' to each side of the equation.
-40 + 15.33333333r + 40 + r2 = 0 + 40

Reorder the terms:
-40 + 40 + 15.33333333r + r2 = 0 + 40

Combine like terms: -40 + 40 = 0
0 + 15.33333333r + r2 = 0 + 40
15.33333333r + r2 = 0 + 40

Combine like terms: 0 + 40 = 40
15.33333333r + r2 = 40

The r term is 15.33333333r.  Take half its coefficient (7.666666665).
Square it (58.77777775) and add it to both sides.

Add '58.77777775' to each side of the equation.
15.33333333r + 58.77777775 + r2 = 40 + 58.77777775

Reorder the terms:
58.77777775 + 15.33333333r + r2 = 40 + 58.77777775

Combine like terms: 40 + 58.77777775 = 98.77777775
58.77777775 + 15.33333333r + r2 = 98.77777775

Factor a perfect square on the left side:
(r + 7.666666665)(r + 7.666666665) = 98.77777775

Calculate the square root of the right side: 9.938701009

Break this problem into two subproblems by setting 
(r + 7.666666665) equal to 9.938701009 and -9.938701009.

Subproblem 1

r + 7.666666665 = 9.938701009 Simplifying r + 7.666666665 = 9.938701009 Reorder the terms: 7.666666665 + r = 9.938701009 Solving 7.666666665 + r = 9.938701009 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-7.666666665' to each side of the equation. 7.666666665 + -7.666666665 + r = 9.938701009 + -7.666666665 Combine like terms: 7.666666665 + -7.666666665 = 0.000000000 0.000000000 + r = 9.938701009 + -7.666666665 r = 9.938701009 + -7.666666665 Combine like terms: 9.938701009 + -7.666666665 = 2.272034344 r = 2.272034344 Simplifying r = 2.272034344

Subproblem 2

r + 7.666666665 = -9.938701009 Simplifying r + 7.666666665 = -9.938701009 Reorder the terms: 7.666666665 + r = -9.938701009 Solving 7.666666665 + r = -9.938701009 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-7.666666665' to each side of the equation. 7.666666665 + -7.666666665 + r = -9.938701009 + -7.666666665 Combine like terms: 7.666666665 + -7.666666665 = 0.000000000 0.000000000 + r = -9.938701009 + -7.666666665 r = -9.938701009 + -7.666666665 Combine like terms: -9.938701009 + -7.666666665 = -17.605367674 r = -17.605367674 Simplifying r = -17.605367674

Solution

The solution to the problem is based on the solutions from the subproblems. r = {2.272034344, -17.605367674}

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