=(3r^2+7r+1)+(4r^2-8r-2)

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Solution for =(3r^2+7r+1)+(4r^2-8r-2) equation:


Simplifying
0 = (3r2 + 7r + 1) + (4r2 + -8r + -2)

Reorder the terms:
0 = (1 + 7r + 3r2) + (4r2 + -8r + -2)

Remove parenthesis around (1 + 7r + 3r2)
0 = 1 + 7r + 3r2 + (4r2 + -8r + -2)

Reorder the terms:
0 = 1 + 7r + 3r2 + (-2 + -8r + 4r2)

Remove parenthesis around (-2 + -8r + 4r2)
0 = 1 + 7r + 3r2 + -2 + -8r + 4r2

Reorder the terms:
0 = 1 + -2 + 7r + -8r + 3r2 + 4r2

Combine like terms: 1 + -2 = -1
0 = -1 + 7r + -8r + 3r2 + 4r2

Combine like terms: 7r + -8r = -1r
0 = -1 + -1r + 3r2 + 4r2

Combine like terms: 3r2 + 4r2 = 7r2
0 = -1 + -1r + 7r2

Solving
0 = -1 + -1r + 7r2

Solving for variable 'r'.

Combine like terms: 0 + 1 = 1
1 + r + -7r2 = -1 + -1r + 7r2 + 1 + r + -7r2

Reorder the terms:
1 + r + -7r2 = -1 + 1 + -1r + r + 7r2 + -7r2

Combine like terms: -1 + 1 = 0
1 + r + -7r2 = 0 + -1r + r + 7r2 + -7r2
1 + r + -7r2 = -1r + r + 7r2 + -7r2

Combine like terms: -1r + r = 0
1 + r + -7r2 = 0 + 7r2 + -7r2
1 + r + -7r2 = 7r2 + -7r2

Combine like terms: 7r2 + -7r2 = 0
1 + r + -7r2 = 0

Begin completing the square.  Divide all terms by
-7 the coefficient of the squared term: 

Divide each side by '-7'.
-0.1428571429 + -0.1428571429r + r2 = 0

Move the constant term to the right:

Add '0.1428571429' to each side of the equation.
-0.1428571429 + -0.1428571429r + 0.1428571429 + r2 = 0 + 0.1428571429

Reorder the terms:
-0.1428571429 + 0.1428571429 + -0.1428571429r + r2 = 0 + 0.1428571429

Combine like terms: -0.1428571429 + 0.1428571429 = 0.0000000000
0.0000000000 + -0.1428571429r + r2 = 0 + 0.1428571429
-0.1428571429r + r2 = 0 + 0.1428571429

Combine like terms: 0 + 0.1428571429 = 0.1428571429
-0.1428571429r + r2 = 0.1428571429

The r term is r.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-0.1428571429r + 0.25 + r2 = 0.1428571429 + 0.25

Reorder the terms:
0.25 + -0.1428571429r + r2 = 0.1428571429 + 0.25

Combine like terms: 0.1428571429 + 0.25 = 0.3928571429
0.25 + -0.1428571429r + r2 = 0.3928571429

Factor a perfect square on the left side:
(r + 0.5)(r + 0.5) = 0.3928571429

Calculate the square root of the right side: 0.626783171

Break this problem into two subproblems by setting 
(r + 0.5) equal to 0.626783171 and -0.626783171.

Subproblem 1

r + 0.5 = 0.626783171 Simplifying r + 0.5 = 0.626783171 Reorder the terms: 0.5 + r = 0.626783171 Solving 0.5 + r = 0.626783171 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + r = 0.626783171 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + r = 0.626783171 + -0.5 r = 0.626783171 + -0.5 Combine like terms: 0.626783171 + -0.5 = 0.126783171 r = 0.126783171 Simplifying r = 0.126783171

Subproblem 2

r + 0.5 = -0.626783171 Simplifying r + 0.5 = -0.626783171 Reorder the terms: 0.5 + r = -0.626783171 Solving 0.5 + r = -0.626783171 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + r = -0.626783171 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + r = -0.626783171 + -0.5 r = -0.626783171 + -0.5 Combine like terms: -0.626783171 + -0.5 = -1.126783171 r = -1.126783171 Simplifying r = -1.126783171

Solution

The solution to the problem is based on the solutions from the subproblems. r = {0.126783171, -1.126783171}

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