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Simplifying 8 = r2 + r Reorder the terms: 8 = r + r2 Solving 8 = r + r2 Solving for variable 'r'. Reorder the terms: 8 + -1r + -1r2 = r + -1r + r2 + -1r2 Combine like terms: r + -1r = 0 8 + -1r + -1r2 = 0 + r2 + -1r2 8 + -1r + -1r2 = r2 + -1r2 Combine like terms: r2 + -1r2 = 0 8 + -1r + -1r2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -8 + r + r2 = 0 Move the constant term to the right: Add '8' to each side of the equation. -8 + r + 8 + r2 = 0 + 8 Reorder the terms: -8 + 8 + r + r2 = 0 + 8 Combine like terms: -8 + 8 = 0 0 + r + r2 = 0 + 8 r + r2 = 0 + 8 Combine like terms: 0 + 8 = 8 r + r2 = 8 The r term is r. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. + 0.25 + r2 = 8 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + r2 = 8 + 0.25 Combine like terms: 8 + 0.25 = 8.25 1.25 + r2 = 8.25 Factor a perfect square on the left side: (r + 0.5)(r + 0.5) = 8.25 Calculate the square root of the right side: 2.872281323 Break this problem into two subproblems by setting (r + 0.5) equal to 2.872281323 and -2.872281323.Subproblem 1
r + 0.5 = 2.872281323 Simplifying r + 0.5 = 2.872281323 Reorder the terms: 0.5 + r = 2.872281323 Solving 0.5 + r = 2.872281323 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + r = 2.872281323 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + r = 2.872281323 + -0.5 r = 2.872281323 + -0.5 Combine like terms: 2.872281323 + -0.5 = 2.372281323 r = 2.372281323 Simplifying r = 2.372281323Subproblem 2
r + 0.5 = -2.872281323 Simplifying r + 0.5 = -2.872281323 Reorder the terms: 0.5 + r = -2.872281323 Solving 0.5 + r = -2.872281323 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + r = -2.872281323 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + r = -2.872281323 + -0.5 r = -2.872281323 + -0.5 Combine like terms: -2.872281323 + -0.5 = -3.372281323 r = -3.372281323 Simplifying r = -3.372281323Solution
The solution to the problem is based on the solutions from the subproblems. r = {2.372281323, -3.372281323}
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