5r^2-55r+123=0

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Solution for 5r^2-55r+123=0 equation:


Simplifying
5r2 + -55r + 123 = 0

Reorder the terms:
123 + -55r + 5r2 = 0

Solving
123 + -55r + 5r2 = 0

Solving for variable 'r'.

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

Divide each side by '5'.
24.6 + -11r + r2 = 0

Move the constant term to the right:

Add '-24.6' to each side of the equation.
24.6 + -11r + -24.6 + r2 = 0 + -24.6

Reorder the terms:
24.6 + -24.6 + -11r + r2 = 0 + -24.6

Combine like terms: 24.6 + -24.6 = 0.0
0.0 + -11r + r2 = 0 + -24.6
-11r + r2 = 0 + -24.6

Combine like terms: 0 + -24.6 = -24.6
-11r + r2 = -24.6

The r term is -11r.  Take half its coefficient (-5.5).
Square it (30.25) and add it to both sides.

Add '30.25' to each side of the equation.
-11r + 30.25 + r2 = -24.6 + 30.25

Reorder the terms:
30.25 + -11r + r2 = -24.6 + 30.25

Combine like terms: -24.6 + 30.25 = 5.65
30.25 + -11r + r2 = 5.65

Factor a perfect square on the left side:
(r + -5.5)(r + -5.5) = 5.65

Calculate the square root of the right side: 2.376972865

Break this problem into two subproblems by setting 
(r + -5.5) equal to 2.376972865 and -2.376972865.

Subproblem 1

r + -5.5 = 2.376972865 Simplifying r + -5.5 = 2.376972865 Reorder the terms: -5.5 + r = 2.376972865 Solving -5.5 + r = 2.376972865 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '5.5' to each side of the equation. -5.5 + 5.5 + r = 2.376972865 + 5.5 Combine like terms: -5.5 + 5.5 = 0.0 0.0 + r = 2.376972865 + 5.5 r = 2.376972865 + 5.5 Combine like terms: 2.376972865 + 5.5 = 7.876972865 r = 7.876972865 Simplifying r = 7.876972865

Subproblem 2

r + -5.5 = -2.376972865 Simplifying r + -5.5 = -2.376972865 Reorder the terms: -5.5 + r = -2.376972865 Solving -5.5 + r = -2.376972865 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '5.5' to each side of the equation. -5.5 + 5.5 + r = -2.376972865 + 5.5 Combine like terms: -5.5 + 5.5 = 0.0 0.0 + r = -2.376972865 + 5.5 r = -2.376972865 + 5.5 Combine like terms: -2.376972865 + 5.5 = 3.123027135 r = 3.123027135 Simplifying r = 3.123027135

Solution

The solution to the problem is based on the solutions from the subproblems. r = {7.876972865, 3.123027135}

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