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Simplifying 9(7 + 5r) * r = 2 Reorder the terms for easier multiplication: 9r(7 + 5r) = 2 (7 * 9r + 5r * 9r) = 2 (63r + 45r2) = 2 Solving 63r + 45r2 = 2 Solving for variable 'r'. Reorder the terms: -2 + 63r + 45r2 = 2 + -2 Combine like terms: 2 + -2 = 0 -2 + 63r + 45r2 = 0 Begin completing the square. Divide all terms by 45 the coefficient of the squared term: Divide each side by '45'. -0.04444444444 + 1.4r + r2 = 0 Move the constant term to the right: Add '0.04444444444' to each side of the equation. -0.04444444444 + 1.4r + 0.04444444444 + r2 = 0 + 0.04444444444 Reorder the terms: -0.04444444444 + 0.04444444444 + 1.4r + r2 = 0 + 0.04444444444 Combine like terms: -0.04444444444 + 0.04444444444 = 0.00000000000 0.00000000000 + 1.4r + r2 = 0 + 0.04444444444 1.4r + r2 = 0 + 0.04444444444 Combine like terms: 0 + 0.04444444444 = 0.04444444444 1.4r + r2 = 0.04444444444 The r term is 1.4r. Take half its coefficient (0.7). Square it (0.49) and add it to both sides. Add '0.49' to each side of the equation. 1.4r + 0.49 + r2 = 0.04444444444 + 0.49 Reorder the terms: 0.49 + 1.4r + r2 = 0.04444444444 + 0.49 Combine like terms: 0.04444444444 + 0.49 = 0.53444444444 0.49 + 1.4r + r2 = 0.53444444444 Factor a perfect square on the left side: (r + 0.7)(r + 0.7) = 0.53444444444 Calculate the square root of the right side: 0.731057073 Break this problem into two subproblems by setting (r + 0.7) equal to 0.731057073 and -0.731057073.Subproblem 1
r + 0.7 = 0.731057073 Simplifying r + 0.7 = 0.731057073 Reorder the terms: 0.7 + r = 0.731057073 Solving 0.7 + r = 0.731057073 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.7' to each side of the equation. 0.7 + -0.7 + r = 0.731057073 + -0.7 Combine like terms: 0.7 + -0.7 = 0.0 0.0 + r = 0.731057073 + -0.7 r = 0.731057073 + -0.7 Combine like terms: 0.731057073 + -0.7 = 0.031057073 r = 0.031057073 Simplifying r = 0.031057073Subproblem 2
r + 0.7 = -0.731057073 Simplifying r + 0.7 = -0.731057073 Reorder the terms: 0.7 + r = -0.731057073 Solving 0.7 + r = -0.731057073 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.7' to each side of the equation. 0.7 + -0.7 + r = -0.731057073 + -0.7 Combine like terms: 0.7 + -0.7 = 0.0 0.0 + r = -0.731057073 + -0.7 r = -0.731057073 + -0.7 Combine like terms: -0.731057073 + -0.7 = -1.431057073 r = -1.431057073 Simplifying r = -1.431057073Solution
The solution to the problem is based on the solutions from the subproblems. r = {0.031057073, -1.431057073}
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