-10=r^2-10r+12

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Solution for -10=r^2-10r+12 equation:


Simplifying
-10 = r2 + -10r + 12

Reorder the terms:
-10 = 12 + -10r + r2

Solving
-10 = 12 + -10r + r2

Solving for variable 'r'.

Combine like terms: -10 + -12 = -22
-22 + 10r + -1r2 = 12 + -10r + r2 + -12 + 10r + -1r2

Reorder the terms:
-22 + 10r + -1r2 = 12 + -12 + -10r + 10r + r2 + -1r2

Combine like terms: 12 + -12 = 0
-22 + 10r + -1r2 = 0 + -10r + 10r + r2 + -1r2
-22 + 10r + -1r2 = -10r + 10r + r2 + -1r2

Combine like terms: -10r + 10r = 0
-22 + 10r + -1r2 = 0 + r2 + -1r2
-22 + 10r + -1r2 = r2 + -1r2

Combine like terms: r2 + -1r2 = 0
-22 + 10r + -1r2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
22 + -10r + r2 = 0

Move the constant term to the right:

Add '-22' to each side of the equation.
22 + -10r + -22 + r2 = 0 + -22

Reorder the terms:
22 + -22 + -10r + r2 = 0 + -22

Combine like terms: 22 + -22 = 0
0 + -10r + r2 = 0 + -22
-10r + r2 = 0 + -22

Combine like terms: 0 + -22 = -22
-10r + r2 = -22

The r term is -10r.  Take half its coefficient (-5).
Square it (25) and add it to both sides.

Add '25' to each side of the equation.
-10r + 25 + r2 = -22 + 25

Reorder the terms:
25 + -10r + r2 = -22 + 25

Combine like terms: -22 + 25 = 3
25 + -10r + r2 = 3

Factor a perfect square on the left side:
(r + -5)(r + -5) = 3

Calculate the square root of the right side: 1.732050808

Break this problem into two subproblems by setting 
(r + -5) equal to 1.732050808 and -1.732050808.

Subproblem 1

r + -5 = 1.732050808 Simplifying r + -5 = 1.732050808 Reorder the terms: -5 + r = 1.732050808 Solving -5 + r = 1.732050808 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + r = 1.732050808 + 5 Combine like terms: -5 + 5 = 0 0 + r = 1.732050808 + 5 r = 1.732050808 + 5 Combine like terms: 1.732050808 + 5 = 6.732050808 r = 6.732050808 Simplifying r = 6.732050808

Subproblem 2

r + -5 = -1.732050808 Simplifying r + -5 = -1.732050808 Reorder the terms: -5 + r = -1.732050808 Solving -5 + r = -1.732050808 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + r = -1.732050808 + 5 Combine like terms: -5 + 5 = 0 0 + r = -1.732050808 + 5 r = -1.732050808 + 5 Combine like terms: -1.732050808 + 5 = 3.267949192 r = 3.267949192 Simplifying r = 3.267949192

Solution

The solution to the problem is based on the solutions from the subproblems. r = {6.732050808, 3.267949192}

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