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