3r^2-42r+110=0

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Solution for 3r^2-42r+110=0 equation:


Simplifying
3r2 + -42r + 110 = 0

Reorder the terms:
110 + -42r + 3r2 = 0

Solving
110 + -42r + 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'.
36.66666667 + -14r + r2 = 0

Move the constant term to the right:

Add '-36.66666667' to each side of the equation.
36.66666667 + -14r + -36.66666667 + r2 = 0 + -36.66666667

Reorder the terms:
36.66666667 + -36.66666667 + -14r + r2 = 0 + -36.66666667

Combine like terms: 36.66666667 + -36.66666667 = 0.00000000
0.00000000 + -14r + r2 = 0 + -36.66666667
-14r + r2 = 0 + -36.66666667

Combine like terms: 0 + -36.66666667 = -36.66666667
-14r + r2 = -36.66666667

The r term is -14r.  Take half its coefficient (-7).
Square it (49) and add it to both sides.

Add '49' to each side of the equation.
-14r + 49 + r2 = -36.66666667 + 49

Reorder the terms:
49 + -14r + r2 = -36.66666667 + 49

Combine like terms: -36.66666667 + 49 = 12.33333333
49 + -14r + r2 = 12.33333333

Factor a perfect square on the left side:
(r + -7)(r + -7) = 12.33333333

Calculate the square root of the right side: 3.511884584

Break this problem into two subproblems by setting 
(r + -7) equal to 3.511884584 and -3.511884584.

Subproblem 1

r + -7 = 3.511884584 Simplifying r + -7 = 3.511884584 Reorder the terms: -7 + r = 3.511884584 Solving -7 + r = 3.511884584 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '7' to each side of the equation. -7 + 7 + r = 3.511884584 + 7 Combine like terms: -7 + 7 = 0 0 + r = 3.511884584 + 7 r = 3.511884584 + 7 Combine like terms: 3.511884584 + 7 = 10.511884584 r = 10.511884584 Simplifying r = 10.511884584

Subproblem 2

r + -7 = -3.511884584 Simplifying r + -7 = -3.511884584 Reorder the terms: -7 + r = -3.511884584 Solving -7 + r = -3.511884584 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '7' to each side of the equation. -7 + 7 + r = -3.511884584 + 7 Combine like terms: -7 + 7 = 0 0 + r = -3.511884584 + 7 r = -3.511884584 + 7 Combine like terms: -3.511884584 + 7 = 3.488115416 r = 3.488115416 Simplifying r = 3.488115416

Solution

The solution to the problem is based on the solutions from the subproblems. r = {10.511884584, 3.488115416}

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