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Simplifying 81 = r2 + 12r + 36 + r2 Reorder the terms: 81 = 36 + 12r + r2 + r2 Combine like terms: r2 + r2 = 2r2 81 = 36 + 12r + 2r2 Solving 81 = 36 + 12r + 2r2 Solving for variable 'r'. Combine like terms: 81 + -36 = 45 45 + -12r + -2r2 = 36 + 12r + 2r2 + -36 + -12r + -2r2 Reorder the terms: 45 + -12r + -2r2 = 36 + -36 + 12r + -12r + 2r2 + -2r2 Combine like terms: 36 + -36 = 0 45 + -12r + -2r2 = 0 + 12r + -12r + 2r2 + -2r2 45 + -12r + -2r2 = 12r + -12r + 2r2 + -2r2 Combine like terms: 12r + -12r = 0 45 + -12r + -2r2 = 0 + 2r2 + -2r2 45 + -12r + -2r2 = 2r2 + -2r2 Combine like terms: 2r2 + -2r2 = 0 45 + -12r + -2r2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -22.5 + 6r + r2 = 0 Move the constant term to the right: Add '22.5' to each side of the equation. -22.5 + 6r + 22.5 + r2 = 0 + 22.5 Reorder the terms: -22.5 + 22.5 + 6r + r2 = 0 + 22.5 Combine like terms: -22.5 + 22.5 = 0.0 0.0 + 6r + r2 = 0 + 22.5 6r + r2 = 0 + 22.5 Combine like terms: 0 + 22.5 = 22.5 6r + r2 = 22.5 The r term is 6r. Take half its coefficient (3). Square it (9) and add it to both sides. Add '9' to each side of the equation. 6r + 9 + r2 = 22.5 + 9 Reorder the terms: 9 + 6r + r2 = 22.5 + 9 Combine like terms: 22.5 + 9 = 31.5 9 + 6r + r2 = 31.5 Factor a perfect square on the left side: (r + 3)(r + 3) = 31.5 Calculate the square root of the right side: 5.61248608 Break this problem into two subproblems by setting (r + 3) equal to 5.61248608 and -5.61248608.Subproblem 1
r + 3 = 5.61248608 Simplifying r + 3 = 5.61248608 Reorder the terms: 3 + r = 5.61248608 Solving 3 + r = 5.61248608 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + r = 5.61248608 + -3 Combine like terms: 3 + -3 = 0 0 + r = 5.61248608 + -3 r = 5.61248608 + -3 Combine like terms: 5.61248608 + -3 = 2.61248608 r = 2.61248608 Simplifying r = 2.61248608Subproblem 2
r + 3 = -5.61248608 Simplifying r + 3 = -5.61248608 Reorder the terms: 3 + r = -5.61248608 Solving 3 + r = -5.61248608 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + r = -5.61248608 + -3 Combine like terms: 3 + -3 = 0 0 + r = -5.61248608 + -3 r = -5.61248608 + -3 Combine like terms: -5.61248608 + -3 = -8.61248608 r = -8.61248608 Simplifying r = -8.61248608Solution
The solution to the problem is based on the solutions from the subproblems. r = {2.61248608, -8.61248608}
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