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Simplifying 10r2 = -9r + 8 Reorder the terms: 10r2 = 8 + -9r Solving 10r2 = 8 + -9r Solving for variable 'r'. Reorder the terms: -8 + 9r + 10r2 = 8 + -9r + -8 + 9r Reorder the terms: -8 + 9r + 10r2 = 8 + -8 + -9r + 9r Combine like terms: 8 + -8 = 0 -8 + 9r + 10r2 = 0 + -9r + 9r -8 + 9r + 10r2 = -9r + 9r Combine like terms: -9r + 9r = 0 -8 + 9r + 10r2 = 0 Begin completing the square. Divide all terms by 10 the coefficient of the squared term: Divide each side by '10'. -0.8 + 0.9r + r2 = 0 Move the constant term to the right: Add '0.8' to each side of the equation. -0.8 + 0.9r + 0.8 + r2 = 0 + 0.8 Reorder the terms: -0.8 + 0.8 + 0.9r + r2 = 0 + 0.8 Combine like terms: -0.8 + 0.8 = 0.0 0.0 + 0.9r + r2 = 0 + 0.8 0.9r + r2 = 0 + 0.8 Combine like terms: 0 + 0.8 = 0.8 0.9r + r2 = 0.8 The r term is 0.9r. Take half its coefficient (0.45). Square it (0.2025) and add it to both sides. Add '0.2025' to each side of the equation. 0.9r + 0.2025 + r2 = 0.8 + 0.2025 Reorder the terms: 0.2025 + 0.9r + r2 = 0.8 + 0.2025 Combine like terms: 0.8 + 0.2025 = 1.0025 0.2025 + 0.9r + r2 = 1.0025 Factor a perfect square on the left side: (r + 0.45)(r + 0.45) = 1.0025 Calculate the square root of the right side: 1.00124922 Break this problem into two subproblems by setting (r + 0.45) equal to 1.00124922 and -1.00124922.Subproblem 1
r + 0.45 = 1.00124922 Simplifying r + 0.45 = 1.00124922 Reorder the terms: 0.45 + r = 1.00124922 Solving 0.45 + r = 1.00124922 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.45' to each side of the equation. 0.45 + -0.45 + r = 1.00124922 + -0.45 Combine like terms: 0.45 + -0.45 = 0.00 0.00 + r = 1.00124922 + -0.45 r = 1.00124922 + -0.45 Combine like terms: 1.00124922 + -0.45 = 0.55124922 r = 0.55124922 Simplifying r = 0.55124922Subproblem 2
r + 0.45 = -1.00124922 Simplifying r + 0.45 = -1.00124922 Reorder the terms: 0.45 + r = -1.00124922 Solving 0.45 + r = -1.00124922 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.45' to each side of the equation. 0.45 + -0.45 + r = -1.00124922 + -0.45 Combine like terms: 0.45 + -0.45 = 0.00 0.00 + r = -1.00124922 + -0.45 r = -1.00124922 + -0.45 Combine like terms: -1.00124922 + -0.45 = -1.45124922 r = -1.45124922 Simplifying r = -1.45124922Solution
The solution to the problem is based on the solutions from the subproblems. r = {0.55124922, -1.45124922}
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