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Simplifying 20r2 = r + 24 Reorder the terms: 20r2 = 24 + r Solving 20r2 = 24 + r Solving for variable 'r'. Reorder the terms: -24 + -1r + 20r2 = 24 + r + -24 + -1r Reorder the terms: -24 + -1r + 20r2 = 24 + -24 + r + -1r Combine like terms: 24 + -24 = 0 -24 + -1r + 20r2 = 0 + r + -1r -24 + -1r + 20r2 = r + -1r Combine like terms: r + -1r = 0 -24 + -1r + 20r2 = 0 Begin completing the square. Divide all terms by 20 the coefficient of the squared term: Divide each side by '20'. -1.2 + -0.05r + r2 = 0 Move the constant term to the right: Add '1.2' to each side of the equation. -1.2 + -0.05r + 1.2 + r2 = 0 + 1.2 Reorder the terms: -1.2 + 1.2 + -0.05r + r2 = 0 + 1.2 Combine like terms: -1.2 + 1.2 = 0.0 0.0 + -0.05r + r2 = 0 + 1.2 -0.05r + r2 = 0 + 1.2 Combine like terms: 0 + 1.2 = 1.2 -0.05r + r2 = 1.2 The r term is -0.05r. Take half its coefficient (-0.025). Square it (0.000625) and add it to both sides. Add '0.000625' to each side of the equation. -0.05r + 0.000625 + r2 = 1.2 + 0.000625 Reorder the terms: 0.000625 + -0.05r + r2 = 1.2 + 0.000625 Combine like terms: 1.2 + 0.000625 = 1.200625 0.000625 + -0.05r + r2 = 1.200625 Factor a perfect square on the left side: (r + -0.025)(r + -0.025) = 1.200625 Calculate the square root of the right side: 1.09573035 Break this problem into two subproblems by setting (r + -0.025) equal to 1.09573035 and -1.09573035.Subproblem 1
r + -0.025 = 1.09573035 Simplifying r + -0.025 = 1.09573035 Reorder the terms: -0.025 + r = 1.09573035 Solving -0.025 + r = 1.09573035 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.025' to each side of the equation. -0.025 + 0.025 + r = 1.09573035 + 0.025 Combine like terms: -0.025 + 0.025 = 0.000 0.000 + r = 1.09573035 + 0.025 r = 1.09573035 + 0.025 Combine like terms: 1.09573035 + 0.025 = 1.12073035 r = 1.12073035 Simplifying r = 1.12073035Subproblem 2
r + -0.025 = -1.09573035 Simplifying r + -0.025 = -1.09573035 Reorder the terms: -0.025 + r = -1.09573035 Solving -0.025 + r = -1.09573035 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.025' to each side of the equation. -0.025 + 0.025 + r = -1.09573035 + 0.025 Combine like terms: -0.025 + 0.025 = 0.000 0.000 + r = -1.09573035 + 0.025 r = -1.09573035 + 0.025 Combine like terms: -1.09573035 + 0.025 = -1.07073035 r = -1.07073035 Simplifying r = -1.07073035Solution
The solution to the problem is based on the solutions from the subproblems. r = {1.12073035, -1.07073035}
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