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Simplifying 23 + 2r = 2r2 Solving 23 + 2r = 2r2 Solving for variable 'r'. Combine like terms: 2r2 + -2r2 = 0 23 + 2r + -2r2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -11.5 + -1r + r2 = 0 Move the constant term to the right: Add '11.5' to each side of the equation. -11.5 + -1r + 11.5 + r2 = 0 + 11.5 Reorder the terms: -11.5 + 11.5 + -1r + r2 = 0 + 11.5 Combine like terms: -11.5 + 11.5 = 0.0 0.0 + -1r + r2 = 0 + 11.5 -1r + r2 = 0 + 11.5 Combine like terms: 0 + 11.5 = 11.5 -1r + r2 = 11.5 The r term is -1r. 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. -1r + 0.25 + r2 = 11.5 + 0.25 Reorder the terms: 0.25 + -1r + r2 = 11.5 + 0.25 Combine like terms: 11.5 + 0.25 = 11.75 0.25 + -1r + r2 = 11.75 Factor a perfect square on the left side: (r + -0.5)(r + -0.5) = 11.75 Calculate the square root of the right side: 3.4278273 Break this problem into two subproblems by setting (r + -0.5) equal to 3.4278273 and -3.4278273.Subproblem 1
r + -0.5 = 3.4278273 Simplifying r + -0.5 = 3.4278273 Reorder the terms: -0.5 + r = 3.4278273 Solving -0.5 + r = 3.4278273 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 = 3.4278273 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + r = 3.4278273 + 0.5 r = 3.4278273 + 0.5 Combine like terms: 3.4278273 + 0.5 = 3.9278273 r = 3.9278273 Simplifying r = 3.9278273Subproblem 2
r + -0.5 = -3.4278273 Simplifying r + -0.5 = -3.4278273 Reorder the terms: -0.5 + r = -3.4278273 Solving -0.5 + r = -3.4278273 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 = -3.4278273 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + r = -3.4278273 + 0.5 r = -3.4278273 + 0.5 Combine like terms: -3.4278273 + 0.5 = -2.9278273 r = -2.9278273 Simplifying r = -2.9278273Solution
The solution to the problem is based on the solutions from the subproblems. r = {3.9278273, -2.9278273}
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