45=r(2r-3)

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Solution for 45=r(2r-3) equation:


Simplifying
45 = r(2r + -3)

Reorder the terms:
45 = r(-3 + 2r)
45 = (-3 * r + 2r * r)
45 = (-3r + 2r2)

Solving
45 = -3r + 2r2

Solving for variable 'r'.

Reorder the terms:
45 + 3r + -2r2 = -3r + 3r + 2r2 + -2r2

Combine like terms: -3r + 3r = 0
45 + 3r + -2r2 = 0 + 2r2 + -2r2
45 + 3r + -2r2 = 2r2 + -2r2

Combine like terms: 2r2 + -2r2 = 0
45 + 3r + -2r2 = 0

Begin completing the square.  Divide all terms by
-2 the coefficient of the squared term: 

Divide each side by '-2'.
-22.5 + -1.5r + r2 = 0

Move the constant term to the right:

Add '22.5' to each side of the equation.
-22.5 + -1.5r + 22.5 + r2 = 0 + 22.5

Reorder the terms:
-22.5 + 22.5 + -1.5r + r2 = 0 + 22.5

Combine like terms: -22.5 + 22.5 = 0.0
0.0 + -1.5r + r2 = 0 + 22.5
-1.5r + r2 = 0 + 22.5

Combine like terms: 0 + 22.5 = 22.5
-1.5r + r2 = 22.5

The r term is -1.5r.  Take half its coefficient (-0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
-1.5r + 0.5625 + r2 = 22.5 + 0.5625

Reorder the terms:
0.5625 + -1.5r + r2 = 22.5 + 0.5625

Combine like terms: 22.5 + 0.5625 = 23.0625
0.5625 + -1.5r + r2 = 23.0625

Factor a perfect square on the left side:
(r + -0.75)(r + -0.75) = 23.0625

Calculate the square root of the right side: 4.802343178

Break this problem into two subproblems by setting 
(r + -0.75) equal to 4.802343178 and -4.802343178.

Subproblem 1

r + -0.75 = 4.802343178 Simplifying r + -0.75 = 4.802343178 Reorder the terms: -0.75 + r = 4.802343178 Solving -0.75 + r = 4.802343178 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + r = 4.802343178 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + r = 4.802343178 + 0.75 r = 4.802343178 + 0.75 Combine like terms: 4.802343178 + 0.75 = 5.552343178 r = 5.552343178 Simplifying r = 5.552343178

Subproblem 2

r + -0.75 = -4.802343178 Simplifying r + -0.75 = -4.802343178 Reorder the terms: -0.75 + r = -4.802343178 Solving -0.75 + r = -4.802343178 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + r = -4.802343178 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + r = -4.802343178 + 0.75 r = -4.802343178 + 0.75 Combine like terms: -4.802343178 + 0.75 = -4.052343178 r = -4.052343178 Simplifying r = -4.052343178

Solution

The solution to the problem is based on the solutions from the subproblems. r = {5.552343178, -4.052343178}

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