0=7r^2+31r-108

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Solution for 0=7r^2+31r-108 equation:


Simplifying
0 = 7r2 + 31r + -108

Reorder the terms:
0 = -108 + 31r + 7r2

Solving
0 = -108 + 31r + 7r2

Solving for variable 'r'.

Combine like terms: 0 + 108 = 108
108 + -31r + -7r2 = -108 + 31r + 7r2 + 108 + -31r + -7r2

Reorder the terms:
108 + -31r + -7r2 = -108 + 108 + 31r + -31r + 7r2 + -7r2

Combine like terms: -108 + 108 = 0
108 + -31r + -7r2 = 0 + 31r + -31r + 7r2 + -7r2
108 + -31r + -7r2 = 31r + -31r + 7r2 + -7r2

Combine like terms: 31r + -31r = 0
108 + -31r + -7r2 = 0 + 7r2 + -7r2
108 + -31r + -7r2 = 7r2 + -7r2

Combine like terms: 7r2 + -7r2 = 0
108 + -31r + -7r2 = 0

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

Divide each side by '-7'.
-15.42857143 + 4.428571429r + r2 = 0

Move the constant term to the right:

Add '15.42857143' to each side of the equation.
-15.42857143 + 4.428571429r + 15.42857143 + r2 = 0 + 15.42857143

Reorder the terms:
-15.42857143 + 15.42857143 + 4.428571429r + r2 = 0 + 15.42857143

Combine like terms: -15.42857143 + 15.42857143 = 0.00000000
0.00000000 + 4.428571429r + r2 = 0 + 15.42857143
4.428571429r + r2 = 0 + 15.42857143

Combine like terms: 0 + 15.42857143 = 15.42857143
4.428571429r + r2 = 15.42857143

The r term is 4.428571429r.  Take half its coefficient (2.214285715).
Square it (4.903061228) and add it to both sides.

Add '4.903061228' to each side of the equation.
4.428571429r + 4.903061228 + r2 = 15.42857143 + 4.903061228

Reorder the terms:
4.903061228 + 4.428571429r + r2 = 15.42857143 + 4.903061228

Combine like terms: 15.42857143 + 4.903061228 = 20.331632658
4.903061228 + 4.428571429r + r2 = 20.331632658

Factor a perfect square on the left side:
(r + 2.214285715)(r + 2.214285715) = 20.331632658

Calculate the square root of the right side: 4.509061173

Break this problem into two subproblems by setting 
(r + 2.214285715) equal to 4.509061173 and -4.509061173.

Subproblem 1

r + 2.214285715 = 4.509061173 Simplifying r + 2.214285715 = 4.509061173 Reorder the terms: 2.214285715 + r = 4.509061173 Solving 2.214285715 + r = 4.509061173 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-2.214285715' to each side of the equation. 2.214285715 + -2.214285715 + r = 4.509061173 + -2.214285715 Combine like terms: 2.214285715 + -2.214285715 = 0.000000000 0.000000000 + r = 4.509061173 + -2.214285715 r = 4.509061173 + -2.214285715 Combine like terms: 4.509061173 + -2.214285715 = 2.294775458 r = 2.294775458 Simplifying r = 2.294775458

Subproblem 2

r + 2.214285715 = -4.509061173 Simplifying r + 2.214285715 = -4.509061173 Reorder the terms: 2.214285715 + r = -4.509061173 Solving 2.214285715 + r = -4.509061173 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-2.214285715' to each side of the equation. 2.214285715 + -2.214285715 + r = -4.509061173 + -2.214285715 Combine like terms: 2.214285715 + -2.214285715 = 0.000000000 0.000000000 + r = -4.509061173 + -2.214285715 r = -4.509061173 + -2.214285715 Combine like terms: -4.509061173 + -2.214285715 = -6.723346888 r = -6.723346888 Simplifying r = -6.723346888

Solution

The solution to the problem is based on the solutions from the subproblems. r = {2.294775458, -6.723346888}

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