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Simplifying -5r(2r + 3) = 3(11 + -4r) + -58 Reorder the terms: -5r(3 + 2r) = 3(11 + -4r) + -58 (3 * -5r + 2r * -5r) = 3(11 + -4r) + -58 (-15r + -10r2) = 3(11 + -4r) + -58 -15r + -10r2 = (11 * 3 + -4r * 3) + -58 -15r + -10r2 = (33 + -12r) + -58 Reorder the terms: -15r + -10r2 = 33 + -58 + -12r Combine like terms: 33 + -58 = -25 -15r + -10r2 = -25 + -12r Solving -15r + -10r2 = -25 + -12r Solving for variable 'r'. Reorder the terms: 25 + -15r + 12r + -10r2 = -25 + -12r + 25 + 12r Combine like terms: -15r + 12r = -3r 25 + -3r + -10r2 = -25 + -12r + 25 + 12r Reorder the terms: 25 + -3r + -10r2 = -25 + 25 + -12r + 12r Combine like terms: -25 + 25 = 0 25 + -3r + -10r2 = 0 + -12r + 12r 25 + -3r + -10r2 = -12r + 12r Combine like terms: -12r + 12r = 0 25 + -3r + -10r2 = 0 Begin completing the square. Divide all terms by -10 the coefficient of the squared term: Divide each side by '-10'. -2.5 + 0.3r + r2 = 0.0 Move the constant term to the right: Add '2.5' to each side of the equation. -2.5 + 0.3r + 2.5 + r2 = 0.0 + 2.5 Reorder the terms: -2.5 + 2.5 + 0.3r + r2 = 0.0 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + 0.3r + r2 = 0.0 + 2.5 0.3r + r2 = 0.0 + 2.5 Combine like terms: 0.0 + 2.5 = 2.5 0.3r + r2 = 2.5 The r term is 0.3r. Take half its coefficient (0.15). Square it (0.0225) and add it to both sides. Add '0.0225' to each side of the equation. 0.3r + 0.0225 + r2 = 2.5 + 0.0225 Reorder the terms: 0.0225 + 0.3r + r2 = 2.5 + 0.0225 Combine like terms: 2.5 + 0.0225 = 2.5225 0.0225 + 0.3r + r2 = 2.5225 Factor a perfect square on the left side: (r + 0.15)(r + 0.15) = 2.5225 Calculate the square root of the right side: 1.588238017 Break this problem into two subproblems by setting (r + 0.15) equal to 1.588238017 and -1.588238017.Subproblem 1
r + 0.15 = 1.588238017 Simplifying r + 0.15 = 1.588238017 Reorder the terms: 0.15 + r = 1.588238017 Solving 0.15 + r = 1.588238017 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.15' to each side of the equation. 0.15 + -0.15 + r = 1.588238017 + -0.15 Combine like terms: 0.15 + -0.15 = 0.00 0.00 + r = 1.588238017 + -0.15 r = 1.588238017 + -0.15 Combine like terms: 1.588238017 + -0.15 = 1.438238017 r = 1.438238017 Simplifying r = 1.438238017Subproblem 2
r + 0.15 = -1.588238017 Simplifying r + 0.15 = -1.588238017 Reorder the terms: 0.15 + r = -1.588238017 Solving 0.15 + r = -1.588238017 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.15' to each side of the equation. 0.15 + -0.15 + r = -1.588238017 + -0.15 Combine like terms: 0.15 + -0.15 = 0.00 0.00 + r = -1.588238017 + -0.15 r = -1.588238017 + -0.15 Combine like terms: -1.588238017 + -0.15 = -1.738238017 r = -1.738238017 Simplifying r = -1.738238017Solution
The solution to the problem is based on the solutions from the subproblems. r = {1.438238017, -1.738238017}
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