5=4r(2r+3)

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Solution for 5=4r(2r+3) equation:


Simplifying
5 = 4r(2r + 3)

Reorder the terms:
5 = 4r(3 + 2r)
5 = (3 * 4r + 2r * 4r)
5 = (12r + 8r2)

Solving
5 = 12r + 8r2

Solving for variable 'r'.

Reorder the terms:
5 + -12r + -8r2 = 12r + -12r + 8r2 + -8r2

Combine like terms: 12r + -12r = 0
5 + -12r + -8r2 = 0 + 8r2 + -8r2
5 + -12r + -8r2 = 8r2 + -8r2

Combine like terms: 8r2 + -8r2 = 0
5 + -12r + -8r2 = 0

Begin completing the square.  Divide all terms by
-8 the coefficient of the squared term: 

Divide each side by '-8'.
-0.625 + 1.5r + r2 = 0

Move the constant term to the right:

Add '0.625' to each side of the equation.
-0.625 + 1.5r + 0.625 + r2 = 0 + 0.625

Reorder the terms:
-0.625 + 0.625 + 1.5r + r2 = 0 + 0.625

Combine like terms: -0.625 + 0.625 = 0.000
0.000 + 1.5r + r2 = 0 + 0.625
1.5r + r2 = 0 + 0.625

Combine like terms: 0 + 0.625 = 0.625
1.5r + r2 = 0.625

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 = 0.625 + 0.5625

Reorder the terms:
0.5625 + 1.5r + r2 = 0.625 + 0.5625

Combine like terms: 0.625 + 0.5625 = 1.1875
0.5625 + 1.5r + r2 = 1.1875

Factor a perfect square on the left side:
(r + 0.75)(r + 0.75) = 1.1875

Calculate the square root of the right side: 1.089724736

Break this problem into two subproblems by setting 
(r + 0.75) equal to 1.089724736 and -1.089724736.

Subproblem 1

r + 0.75 = 1.089724736 Simplifying r + 0.75 = 1.089724736 Reorder the terms: 0.75 + r = 1.089724736 Solving 0.75 + r = 1.089724736 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 = 1.089724736 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + r = 1.089724736 + -0.75 r = 1.089724736 + -0.75 Combine like terms: 1.089724736 + -0.75 = 0.339724736 r = 0.339724736 Simplifying r = 0.339724736

Subproblem 2

r + 0.75 = -1.089724736 Simplifying r + 0.75 = -1.089724736 Reorder the terms: 0.75 + r = -1.089724736 Solving 0.75 + r = -1.089724736 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 = -1.089724736 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + r = -1.089724736 + -0.75 r = -1.089724736 + -0.75 Combine like terms: -1.089724736 + -0.75 = -1.839724736 r = -1.839724736 Simplifying r = -1.839724736

Solution

The solution to the problem is based on the solutions from the subproblems. r = {0.339724736, -1.839724736}

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