5r(2r-5)=5r-9

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Solution for 5r(2r-5)=5r-9 equation:


Simplifying
5r(2r + -5) = 5r + -9

Reorder the terms:
5r(-5 + 2r) = 5r + -9
(-5 * 5r + 2r * 5r) = 5r + -9
(-25r + 10r2) = 5r + -9

Reorder the terms:
-25r + 10r2 = -9 + 5r

Solving
-25r + 10r2 = -9 + 5r

Solving for variable 'r'.

Reorder the terms:
9 + -25r + -5r + 10r2 = -9 + 5r + 9 + -5r

Combine like terms: -25r + -5r = -30r
9 + -30r + 10r2 = -9 + 5r + 9 + -5r

Reorder the terms:
9 + -30r + 10r2 = -9 + 9 + 5r + -5r

Combine like terms: -9 + 9 = 0
9 + -30r + 10r2 = 0 + 5r + -5r
9 + -30r + 10r2 = 5r + -5r

Combine like terms: 5r + -5r = 0
9 + -30r + 10r2 = 0

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

Divide each side by '10'.
0.9 + -3r + r2 = 0.0

Move the constant term to the right:

Add '-0.9' to each side of the equation.
0.9 + -3r + -0.9 + r2 = 0.0 + -0.9

Reorder the terms:
0.9 + -0.9 + -3r + r2 = 0.0 + -0.9

Combine like terms: 0.9 + -0.9 = 0.0
0.0 + -3r + r2 = 0.0 + -0.9
-3r + r2 = 0.0 + -0.9

Combine like terms: 0.0 + -0.9 = -0.9
-3r + r2 = -0.9

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 = -0.9 + 2.25

Reorder the terms:
2.25 + -3r + r2 = -0.9 + 2.25

Combine like terms: -0.9 + 2.25 = 1.35
2.25 + -3r + r2 = 1.35

Factor a perfect square on the left side:
(r + -1.5)(r + -1.5) = 1.35

Calculate the square root of the right side: 1.161895004

Break this problem into two subproblems by setting 
(r + -1.5) equal to 1.161895004 and -1.161895004.

Subproblem 1

r + -1.5 = 1.161895004 Simplifying r + -1.5 = 1.161895004 Reorder the terms: -1.5 + r = 1.161895004 Solving -1.5 + r = 1.161895004 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 = 1.161895004 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + r = 1.161895004 + 1.5 r = 1.161895004 + 1.5 Combine like terms: 1.161895004 + 1.5 = 2.661895004 r = 2.661895004 Simplifying r = 2.661895004

Subproblem 2

r + -1.5 = -1.161895004 Simplifying r + -1.5 = -1.161895004 Reorder the terms: -1.5 + r = -1.161895004 Solving -1.5 + r = -1.161895004 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 = -1.161895004 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + r = -1.161895004 + 1.5 r = -1.161895004 + 1.5 Combine like terms: -1.161895004 + 1.5 = 0.338104996 r = 0.338104996 Simplifying r = 0.338104996

Solution

The solution to the problem is based on the solutions from the subproblems. r = {2.661895004, 0.338104996}

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