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Simplifying r = 4 + -9r2 Solving r = 4 + -9r2 Solving for variable 'r'. Reorder the terms: -4 + r + 9r2 = 4 + -9r2 + -4 + 9r2 Reorder the terms: -4 + r + 9r2 = 4 + -4 + -9r2 + 9r2 Combine like terms: 4 + -4 = 0 -4 + r + 9r2 = 0 + -9r2 + 9r2 -4 + r + 9r2 = -9r2 + 9r2 Combine like terms: -9r2 + 9r2 = 0 -4 + r + 9r2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -0.4444444444 + 0.1111111111r + r2 = 0 Move the constant term to the right: Add '0.4444444444' to each side of the equation. -0.4444444444 + 0.1111111111r + 0.4444444444 + r2 = 0 + 0.4444444444 Reorder the terms: -0.4444444444 + 0.4444444444 + 0.1111111111r + r2 = 0 + 0.4444444444 Combine like terms: -0.4444444444 + 0.4444444444 = 0.0000000000 0.0000000000 + 0.1111111111r + r2 = 0 + 0.4444444444 0.1111111111r + r2 = 0 + 0.4444444444 Combine like terms: 0 + 0.4444444444 = 0.4444444444 0.1111111111r + r2 = 0.4444444444 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. 0.1111111111r + 0.25 + r2 = 0.4444444444 + 0.25 Reorder the terms: 0.25 + 0.1111111111r + r2 = 0.4444444444 + 0.25 Combine like terms: 0.4444444444 + 0.25 = 0.6944444444 0.25 + 0.1111111111r + r2 = 0.6944444444 Factor a perfect square on the left side: (r + 0.5)(r + 0.5) = 0.6944444444 Calculate the square root of the right side: 0.833333333 Break this problem into two subproblems by setting (r + 0.5) equal to 0.833333333 and -0.833333333.Subproblem 1
r + 0.5 = 0.833333333 Simplifying r + 0.5 = 0.833333333 Reorder the terms: 0.5 + r = 0.833333333 Solving 0.5 + r = 0.833333333 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 = 0.833333333 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + r = 0.833333333 + -0.5 r = 0.833333333 + -0.5 Combine like terms: 0.833333333 + -0.5 = 0.333333333 r = 0.333333333 Simplifying r = 0.333333333Subproblem 2
r + 0.5 = -0.833333333 Simplifying r + 0.5 = -0.833333333 Reorder the terms: 0.5 + r = -0.833333333 Solving 0.5 + r = -0.833333333 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 = -0.833333333 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + r = -0.833333333 + -0.5 r = -0.833333333 + -0.5 Combine like terms: -0.833333333 + -0.5 = -1.333333333 r = -1.333333333 Simplifying r = -1.333333333Solution
The solution to the problem is based on the solutions from the subproblems. r = {0.333333333, -1.333333333}
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