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Simplifying 0 = 2.135795464r2 + 2r + -0.135795464 Reorder the terms: 0 = -0.135795464 + 2r + 2.135795464r2 Solving 0 = -0.135795464 + 2r + 2.135795464r2 Solving for variable 'r'. Combine like terms: 0 + 0.135795464 = 0.135795464 0.135795464 + -2r + -2.135795464r2 = -0.135795464 + 2r + 2.135795464r2 + 0.135795464 + -2r + -2.135795464r2 Reorder the terms: 0.135795464 + -2r + -2.135795464r2 = -0.135795464 + 0.135795464 + 2r + -2r + 2.135795464r2 + -2.135795464r2 Combine like terms: -0.135795464 + 0.135795464 = 0.000000000 0.135795464 + -2r + -2.135795464r2 = 0.000000000 + 2r + -2r + 2.135795464r2 + -2.135795464r2 0.135795464 + -2r + -2.135795464r2 = 2r + -2r + 2.135795464r2 + -2.135795464r2 Combine like terms: 2r + -2r = 0 0.135795464 + -2r + -2.135795464r2 = 0 + 2.135795464r2 + -2.135795464r2 0.135795464 + -2r + -2.135795464r2 = 2.135795464r2 + -2.135795464r2 Combine like terms: 2.135795464r2 + -2.135795464r2 = 0.000000000 0.135795464 + -2r + -2.135795464r2 = 0.000000000 Begin completing the square. Divide all terms by -2.135795464 the coefficient of the squared term: Divide each side by '-2.135795464'. -0.0635807437 + 0.9364192563r + r2 = 0 Move the constant term to the right: Add '0.0635807437' to each side of the equation. -0.0635807437 + 0.9364192563r + 0.0635807437 + r2 = 0 + 0.0635807437 Reorder the terms: -0.0635807437 + 0.0635807437 + 0.9364192563r + r2 = 0 + 0.0635807437 Combine like terms: -0.0635807437 + 0.0635807437 = 0.0000000000 0.0000000000 + 0.9364192563r + r2 = 0 + 0.0635807437 0.9364192563r + r2 = 0 + 0.0635807437 Combine like terms: 0 + 0.0635807437 = 0.0635807437 0.9364192563r + r2 = 0.0635807437 The r term is 0.9364192563r. Take half its coefficient (0.4682096282). Square it (0.2192202559) and add it to both sides. Add '0.2192202559' to each side of the equation. 0.9364192563r + 0.2192202559 + r2 = 0.0635807437 + 0.2192202559 Reorder the terms: 0.2192202559 + 0.9364192563r + r2 = 0.0635807437 + 0.2192202559 Combine like terms: 0.0635807437 + 0.2192202559 = 0.2828009996 0.2192202559 + 0.9364192563r + r2 = 0.2828009996 Factor a perfect square on the left side: (r + 0.4682096282)(r + 0.4682096282) = 0.2828009996 Calculate the square root of the right side: 0.531790372 Break this problem into two subproblems by setting (r + 0.4682096282) equal to 0.531790372 and -0.531790372.Subproblem 1
r + 0.4682096282 = 0.531790372 Simplifying r + 0.4682096282 = 0.531790372 Reorder the terms: 0.4682096282 + r = 0.531790372 Solving 0.4682096282 + r = 0.531790372 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.4682096282' to each side of the equation. 0.4682096282 + -0.4682096282 + r = 0.531790372 + -0.4682096282 Combine like terms: 0.4682096282 + -0.4682096282 = 0.0000000000 0.0000000000 + r = 0.531790372 + -0.4682096282 r = 0.531790372 + -0.4682096282 Combine like terms: 0.531790372 + -0.4682096282 = 0.0635807438 r = 0.0635807438 Simplifying r = 0.0635807438Subproblem 2
r + 0.4682096282 = -0.531790372 Simplifying r + 0.4682096282 = -0.531790372 Reorder the terms: 0.4682096282 + r = -0.531790372 Solving 0.4682096282 + r = -0.531790372 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.4682096282' to each side of the equation. 0.4682096282 + -0.4682096282 + r = -0.531790372 + -0.4682096282 Combine like terms: 0.4682096282 + -0.4682096282 = 0.0000000000 0.0000000000 + r = -0.531790372 + -0.4682096282 r = -0.531790372 + -0.4682096282 Combine like terms: -0.531790372 + -0.4682096282 = -1.0000000002 r = -1.0000000002 Simplifying r = -1.0000000002Solution
The solution to the problem is based on the solutions from the subproblems. r = {0.0635807438, -1.0000000002}
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