-2r(r+3)=-23

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Solution for -2r(r+3)=-23 equation:


Simplifying
-2r(r + 3) = -23

Reorder the terms:
-2r(3 + r) = -23
(3 * -2r + r * -2r) = -23
(-6r + -2r2) = -23

Solving
-6r + -2r2 = -23

Solving for variable 'r'.

Reorder the terms:
23 + -6r + -2r2 = -23 + 23

Combine like terms: -23 + 23 = 0
23 + -6r + -2r2 = 0

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

Divide each side by '-2'.
-11.5 + 3r + r2 = 0

Move the constant term to the right:

Add '11.5' to each side of the equation.
-11.5 + 3r + 11.5 + r2 = 0 + 11.5

Reorder the terms:
-11.5 + 11.5 + 3r + r2 = 0 + 11.5

Combine like terms: -11.5 + 11.5 = 0.0
0.0 + 3r + r2 = 0 + 11.5
3r + r2 = 0 + 11.5

Combine like terms: 0 + 11.5 = 11.5
3r + r2 = 11.5

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 = 11.5 + 2.25

Reorder the terms:
2.25 + 3r + r2 = 11.5 + 2.25

Combine like terms: 11.5 + 2.25 = 13.75
2.25 + 3r + r2 = 13.75

Factor a perfect square on the left side:
(r + 1.5)(r + 1.5) = 13.75

Calculate the square root of the right side: 3.708099244

Break this problem into two subproblems by setting 
(r + 1.5) equal to 3.708099244 and -3.708099244.

Subproblem 1

r + 1.5 = 3.708099244 Simplifying r + 1.5 = 3.708099244 Reorder the terms: 1.5 + r = 3.708099244 Solving 1.5 + r = 3.708099244 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 = 3.708099244 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + r = 3.708099244 + -1.5 r = 3.708099244 + -1.5 Combine like terms: 3.708099244 + -1.5 = 2.208099244 r = 2.208099244 Simplifying r = 2.208099244

Subproblem 2

r + 1.5 = -3.708099244 Simplifying r + 1.5 = -3.708099244 Reorder the terms: 1.5 + r = -3.708099244 Solving 1.5 + r = -3.708099244 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 = -3.708099244 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + r = -3.708099244 + -1.5 r = -3.708099244 + -1.5 Combine like terms: -3.708099244 + -1.5 = -5.208099244 r = -5.208099244 Simplifying r = -5.208099244

Solution

The solution to the problem is based on the solutions from the subproblems. r = {2.208099244, -5.208099244}

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