(8r-1)(5+2r)=(4r+5)2

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Solution for (8r-1)(5+2r)=(4r+5)2 equation:


Simplifying
(8r + -1)(5 + 2r) = (4r + 5) * 2

Reorder the terms:
(-1 + 8r)(5 + 2r) = (4r + 5) * 2

Multiply (-1 + 8r) * (5 + 2r)
(-1(5 + 2r) + 8r * (5 + 2r)) = (4r + 5) * 2
((5 * -1 + 2r * -1) + 8r * (5 + 2r)) = (4r + 5) * 2
((-5 + -2r) + 8r * (5 + 2r)) = (4r + 5) * 2
(-5 + -2r + (5 * 8r + 2r * 8r)) = (4r + 5) * 2
(-5 + -2r + (40r + 16r2)) = (4r + 5) * 2

Combine like terms: -2r + 40r = 38r
(-5 + 38r + 16r2) = (4r + 5) * 2

Reorder the terms:
-5 + 38r + 16r2 = (5 + 4r) * 2

Reorder the terms for easier multiplication:
-5 + 38r + 16r2 = 2(5 + 4r)
-5 + 38r + 16r2 = (5 * 2 + 4r * 2)
-5 + 38r + 16r2 = (10 + 8r)

Solving
-5 + 38r + 16r2 = 10 + 8r

Solving for variable 'r'.

Reorder the terms:
-5 + -10 + 38r + -8r + 16r2 = 10 + 8r + -10 + -8r

Combine like terms: -5 + -10 = -15
-15 + 38r + -8r + 16r2 = 10 + 8r + -10 + -8r

Combine like terms: 38r + -8r = 30r
-15 + 30r + 16r2 = 10 + 8r + -10 + -8r

Reorder the terms:
-15 + 30r + 16r2 = 10 + -10 + 8r + -8r

Combine like terms: 10 + -10 = 0
-15 + 30r + 16r2 = 0 + 8r + -8r
-15 + 30r + 16r2 = 8r + -8r

Combine like terms: 8r + -8r = 0
-15 + 30r + 16r2 = 0

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

Divide each side by '16'.
-0.9375 + 1.875r + r2 = 0

Move the constant term to the right:

Add '0.9375' to each side of the equation.
-0.9375 + 1.875r + 0.9375 + r2 = 0 + 0.9375

Reorder the terms:
-0.9375 + 0.9375 + 1.875r + r2 = 0 + 0.9375

Combine like terms: -0.9375 + 0.9375 = 0.0000
0.0000 + 1.875r + r2 = 0 + 0.9375
1.875r + r2 = 0 + 0.9375

Combine like terms: 0 + 0.9375 = 0.9375
1.875r + r2 = 0.9375

The r term is 1.875r.  Take half its coefficient (0.9375).
Square it (0.87890625) and add it to both sides.

Add '0.87890625' to each side of the equation.
1.875r + 0.87890625 + r2 = 0.9375 + 0.87890625

Reorder the terms:
0.87890625 + 1.875r + r2 = 0.9375 + 0.87890625

Combine like terms: 0.9375 + 0.87890625 = 1.81640625
0.87890625 + 1.875r + r2 = 1.81640625

Factor a perfect square on the left side:
(r + 0.9375)(r + 0.9375) = 1.81640625

Calculate the square root of the right side: 1.347741166

Break this problem into two subproblems by setting 
(r + 0.9375) equal to 1.347741166 and -1.347741166.

Subproblem 1

r + 0.9375 = 1.347741166 Simplifying r + 0.9375 = 1.347741166 Reorder the terms: 0.9375 + r = 1.347741166 Solving 0.9375 + r = 1.347741166 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.9375' to each side of the equation. 0.9375 + -0.9375 + r = 1.347741166 + -0.9375 Combine like terms: 0.9375 + -0.9375 = 0.0000 0.0000 + r = 1.347741166 + -0.9375 r = 1.347741166 + -0.9375 Combine like terms: 1.347741166 + -0.9375 = 0.410241166 r = 0.410241166 Simplifying r = 0.410241166

Subproblem 2

r + 0.9375 = -1.347741166 Simplifying r + 0.9375 = -1.347741166 Reorder the terms: 0.9375 + r = -1.347741166 Solving 0.9375 + r = -1.347741166 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.9375' to each side of the equation. 0.9375 + -0.9375 + r = -1.347741166 + -0.9375 Combine like terms: 0.9375 + -0.9375 = 0.0000 0.0000 + r = -1.347741166 + -0.9375 r = -1.347741166 + -0.9375 Combine like terms: -1.347741166 + -0.9375 = -2.285241166 r = -2.285241166 Simplifying r = -2.285241166

Solution

The solution to the problem is based on the solutions from the subproblems. r = {0.410241166, -2.285241166}

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