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Simplifying 10r2 + -21r = -4 + -6 Reorder the terms: -21r + 10r2 = -4 + -6 Combine like terms: -4 + -6 = -10 -21r + 10r2 = -10 Solving -21r + 10r2 = -10 Solving for variable 'r'. Reorder the terms: 10 + -21r + 10r2 = -10 + 10 Combine like terms: -10 + 10 = 0 10 + -21r + 10r2 = 0 Begin completing the square. Divide all terms by 10 the coefficient of the squared term: Divide each side by '10'. 1 + -2.1r + r2 = 0 Move the constant term to the right: Add '-1' to each side of the equation. 1 + -2.1r + -1 + r2 = 0 + -1 Reorder the terms: 1 + -1 + -2.1r + r2 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + -2.1r + r2 = 0 + -1 -2.1r + r2 = 0 + -1 Combine like terms: 0 + -1 = -1 -2.1r + r2 = -1 The r term is -2.1r. Take half its coefficient (-1.05). Square it (1.1025) and add it to both sides. Add '1.1025' to each side of the equation. -2.1r + 1.1025 + r2 = -1 + 1.1025 Reorder the terms: 1.1025 + -2.1r + r2 = -1 + 1.1025 Combine like terms: -1 + 1.1025 = 0.1025 1.1025 + -2.1r + r2 = 0.1025 Factor a perfect square on the left side: (r + -1.05)(r + -1.05) = 0.1025 Calculate the square root of the right side: 0.320156212 Break this problem into two subproblems by setting (r + -1.05) equal to 0.320156212 and -0.320156212.Subproblem 1
r + -1.05 = 0.320156212 Simplifying r + -1.05 = 0.320156212 Reorder the terms: -1.05 + r = 0.320156212 Solving -1.05 + r = 0.320156212 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1.05' to each side of the equation. -1.05 + 1.05 + r = 0.320156212 + 1.05 Combine like terms: -1.05 + 1.05 = 0.00 0.00 + r = 0.320156212 + 1.05 r = 0.320156212 + 1.05 Combine like terms: 0.320156212 + 1.05 = 1.370156212 r = 1.370156212 Simplifying r = 1.370156212Subproblem 2
r + -1.05 = -0.320156212 Simplifying r + -1.05 = -0.320156212 Reorder the terms: -1.05 + r = -0.320156212 Solving -1.05 + r = -0.320156212 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1.05' to each side of the equation. -1.05 + 1.05 + r = -0.320156212 + 1.05 Combine like terms: -1.05 + 1.05 = 0.00 0.00 + r = -0.320156212 + 1.05 r = -0.320156212 + 1.05 Combine like terms: -0.320156212 + 1.05 = 0.729843788 r = 0.729843788 Simplifying r = 0.729843788Solution
The solution to the problem is based on the solutions from the subproblems. r = {1.370156212, 0.729843788}
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