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Simplifying 5r2 + 24r = -14 Reorder the terms: 24r + 5r2 = -14 Solving 24r + 5r2 = -14 Solving for variable 'r'. Reorder the terms: 14 + 24r + 5r2 = -14 + 14 Combine like terms: -14 + 14 = 0 14 + 24r + 5r2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. 2.8 + 4.8r + r2 = 0 Move the constant term to the right: Add '-2.8' to each side of the equation. 2.8 + 4.8r + -2.8 + r2 = 0 + -2.8 Reorder the terms: 2.8 + -2.8 + 4.8r + r2 = 0 + -2.8 Combine like terms: 2.8 + -2.8 = 0.0 0.0 + 4.8r + r2 = 0 + -2.8 4.8r + r2 = 0 + -2.8 Combine like terms: 0 + -2.8 = -2.8 4.8r + r2 = -2.8 The r term is 4.8r. Take half its coefficient (2.4). Square it (5.76) and add it to both sides. Add '5.76' to each side of the equation. 4.8r + 5.76 + r2 = -2.8 + 5.76 Reorder the terms: 5.76 + 4.8r + r2 = -2.8 + 5.76 Combine like terms: -2.8 + 5.76 = 2.96 5.76 + 4.8r + r2 = 2.96 Factor a perfect square on the left side: (r + 2.4)(r + 2.4) = 2.96 Calculate the square root of the right side: 1.720465053 Break this problem into two subproblems by setting (r + 2.4) equal to 1.720465053 and -1.720465053.Subproblem 1
r + 2.4 = 1.720465053 Simplifying r + 2.4 = 1.720465053 Reorder the terms: 2.4 + r = 1.720465053 Solving 2.4 + r = 1.720465053 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-2.4' to each side of the equation. 2.4 + -2.4 + r = 1.720465053 + -2.4 Combine like terms: 2.4 + -2.4 = 0.0 0.0 + r = 1.720465053 + -2.4 r = 1.720465053 + -2.4 Combine like terms: 1.720465053 + -2.4 = -0.679534947 r = -0.679534947 Simplifying r = -0.679534947Subproblem 2
r + 2.4 = -1.720465053 Simplifying r + 2.4 = -1.720465053 Reorder the terms: 2.4 + r = -1.720465053 Solving 2.4 + r = -1.720465053 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-2.4' to each side of the equation. 2.4 + -2.4 + r = -1.720465053 + -2.4 Combine like terms: 2.4 + -2.4 = 0.0 0.0 + r = -1.720465053 + -2.4 r = -1.720465053 + -2.4 Combine like terms: -1.720465053 + -2.4 = -4.120465053 r = -4.120465053 Simplifying r = -4.120465053Solution
The solution to the problem is based on the solutions from the subproblems. r = {-0.679534947, -4.120465053}
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