3r^2-9r=5

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Solution for 3r^2-9r=5 equation:


Simplifying
3r2 + -9r = 5

Reorder the terms:
-9r + 3r2 = 5

Solving
-9r + 3r2 = 5

Solving for variable 'r'.

Reorder the terms:
-5 + -9r + 3r2 = 5 + -5

Combine like terms: 5 + -5 = 0
-5 + -9r + 3r2 = 0

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

Divide each side by '3'.
-1.666666667 + -3r + r2 = 0

Move the constant term to the right:

Add '1.666666667' to each side of the equation.
-1.666666667 + -3r + 1.666666667 + r2 = 0 + 1.666666667

Reorder the terms:
-1.666666667 + 1.666666667 + -3r + r2 = 0 + 1.666666667

Combine like terms: -1.666666667 + 1.666666667 = 0.000000000
0.000000000 + -3r + r2 = 0 + 1.666666667
-3r + r2 = 0 + 1.666666667

Combine like terms: 0 + 1.666666667 = 1.666666667
-3r + r2 = 1.666666667

The r term is -3r.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3r + 2.25 + r2 = 1.666666667 + 2.25

Reorder the terms:
2.25 + -3r + r2 = 1.666666667 + 2.25

Combine like terms: 1.666666667 + 2.25 = 3.916666667
2.25 + -3r + r2 = 3.916666667

Factor a perfect square on the left side:
(r + -1.5)(r + -1.5) = 3.916666667

Calculate the square root of the right side: 1.979057015

Break this problem into two subproblems by setting 
(r + -1.5) equal to 1.979057015 and -1.979057015.

Subproblem 1

r + -1.5 = 1.979057015 Simplifying r + -1.5 = 1.979057015 Reorder the terms: -1.5 + r = 1.979057015 Solving -1.5 + r = 1.979057015 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + r = 1.979057015 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + r = 1.979057015 + 1.5 r = 1.979057015 + 1.5 Combine like terms: 1.979057015 + 1.5 = 3.479057015 r = 3.479057015 Simplifying r = 3.479057015

Subproblem 2

r + -1.5 = -1.979057015 Simplifying r + -1.5 = -1.979057015 Reorder the terms: -1.5 + r = -1.979057015 Solving -1.5 + r = -1.979057015 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + r = -1.979057015 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + r = -1.979057015 + 1.5 r = -1.979057015 + 1.5 Combine like terms: -1.979057015 + 1.5 = -0.479057015 r = -0.479057015 Simplifying r = -0.479057015

Solution

The solution to the problem is based on the solutions from the subproblems. r = {3.479057015, -0.479057015}

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