4r(2r-2)+7(4+7r)=

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Solution for 4r(2r-2)+7(4+7r)= equation:


Simplifying
4r(2r + -2) + 7(4 + 7r) = 0

Reorder the terms:
4r(-2 + 2r) + 7(4 + 7r) = 0
(-2 * 4r + 2r * 4r) + 7(4 + 7r) = 0
(-8r + 8r2) + 7(4 + 7r) = 0
-8r + 8r2 + (4 * 7 + 7r * 7) = 0
-8r + 8r2 + (28 + 49r) = 0

Reorder the terms:
28 + -8r + 49r + 8r2 = 0

Combine like terms: -8r + 49r = 41r
28 + 41r + 8r2 = 0

Solving
28 + 41r + 8r2 = 0

Solving for variable 'r'.

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

Divide each side by '8'.
3.5 + 5.125r + r2 = 0

Move the constant term to the right:

Add '-3.5' to each side of the equation.
3.5 + 5.125r + -3.5 + r2 = 0 + -3.5

Reorder the terms:
3.5 + -3.5 + 5.125r + r2 = 0 + -3.5

Combine like terms: 3.5 + -3.5 = 0.0
0.0 + 5.125r + r2 = 0 + -3.5
5.125r + r2 = 0 + -3.5

Combine like terms: 0 + -3.5 = -3.5
5.125r + r2 = -3.5

The r term is 5.125r.  Take half its coefficient (2.5625).
Square it (6.56640625) and add it to both sides.

Add '6.56640625' to each side of the equation.
5.125r + 6.56640625 + r2 = -3.5 + 6.56640625

Reorder the terms:
6.56640625 + 5.125r + r2 = -3.5 + 6.56640625

Combine like terms: -3.5 + 6.56640625 = 3.06640625
6.56640625 + 5.125r + r2 = 3.06640625

Factor a perfect square on the left side:
(r + 2.5625)(r + 2.5625) = 3.06640625

Calculate the square root of the right side: 1.751115716

Break this problem into two subproblems by setting 
(r + 2.5625) equal to 1.751115716 and -1.751115716.

Subproblem 1

r + 2.5625 = 1.751115716 Simplifying r + 2.5625 = 1.751115716 Reorder the terms: 2.5625 + r = 1.751115716 Solving 2.5625 + r = 1.751115716 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-2.5625' to each side of the equation. 2.5625 + -2.5625 + r = 1.751115716 + -2.5625 Combine like terms: 2.5625 + -2.5625 = 0.0000 0.0000 + r = 1.751115716 + -2.5625 r = 1.751115716 + -2.5625 Combine like terms: 1.751115716 + -2.5625 = -0.811384284 r = -0.811384284 Simplifying r = -0.811384284

Subproblem 2

r + 2.5625 = -1.751115716 Simplifying r + 2.5625 = -1.751115716 Reorder the terms: 2.5625 + r = -1.751115716 Solving 2.5625 + r = -1.751115716 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-2.5625' to each side of the equation. 2.5625 + -2.5625 + r = -1.751115716 + -2.5625 Combine like terms: 2.5625 + -2.5625 = 0.0000 0.0000 + r = -1.751115716 + -2.5625 r = -1.751115716 + -2.5625 Combine like terms: -1.751115716 + -2.5625 = -4.313615716 r = -4.313615716 Simplifying r = -4.313615716

Solution

The solution to the problem is based on the solutions from the subproblems. r = {-0.811384284, -4.313615716}

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