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Simplifying 9r2 + -18r + -27 = -4 Reorder the terms: -27 + -18r + 9r2 = -4 Solving -27 + -18r + 9r2 = -4 Solving for variable 'r'. Reorder the terms: -27 + 4 + -18r + 9r2 = -4 + 4 Combine like terms: -27 + 4 = -23 -23 + -18r + 9r2 = -4 + 4 Combine like terms: -4 + 4 = 0 -23 + -18r + 9r2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -2.555555556 + -2r + r2 = 0 Move the constant term to the right: Add '2.555555556' to each side of the equation. -2.555555556 + -2r + 2.555555556 + r2 = 0 + 2.555555556 Reorder the terms: -2.555555556 + 2.555555556 + -2r + r2 = 0 + 2.555555556 Combine like terms: -2.555555556 + 2.555555556 = 0.000000000 0.000000000 + -2r + r2 = 0 + 2.555555556 -2r + r2 = 0 + 2.555555556 Combine like terms: 0 + 2.555555556 = 2.555555556 -2r + r2 = 2.555555556 The r term is -2r. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2r + 1 + r2 = 2.555555556 + 1 Reorder the terms: 1 + -2r + r2 = 2.555555556 + 1 Combine like terms: 2.555555556 + 1 = 3.555555556 1 + -2r + r2 = 3.555555556 Factor a perfect square on the left side: (r + -1)(r + -1) = 3.555555556 Calculate the square root of the right side: 1.885618083 Break this problem into two subproblems by setting (r + -1) equal to 1.885618083 and -1.885618083.Subproblem 1
r + -1 = 1.885618083 Simplifying r + -1 = 1.885618083 Reorder the terms: -1 + r = 1.885618083 Solving -1 + r = 1.885618083 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + r = 1.885618083 + 1 Combine like terms: -1 + 1 = 0 0 + r = 1.885618083 + 1 r = 1.885618083 + 1 Combine like terms: 1.885618083 + 1 = 2.885618083 r = 2.885618083 Simplifying r = 2.885618083Subproblem 2
r + -1 = -1.885618083 Simplifying r + -1 = -1.885618083 Reorder the terms: -1 + r = -1.885618083 Solving -1 + r = -1.885618083 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + r = -1.885618083 + 1 Combine like terms: -1 + 1 = 0 0 + r = -1.885618083 + 1 r = -1.885618083 + 1 Combine like terms: -1.885618083 + 1 = -0.885618083 r = -0.885618083 Simplifying r = -0.885618083Solution
The solution to the problem is based on the solutions from the subproblems. r = {2.885618083, -0.885618083}
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