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Simplifying 13r2 + -5r + -21 = 0 Reorder the terms: -21 + -5r + 13r2 = 0 Solving -21 + -5r + 13r2 = 0 Solving for variable 'r'. Begin completing the square. Divide all terms by 13 the coefficient of the squared term: Divide each side by '13'. -1.615384615 + -0.3846153846r + r2 = 0 Move the constant term to the right: Add '1.615384615' to each side of the equation. -1.615384615 + -0.3846153846r + 1.615384615 + r2 = 0 + 1.615384615 Reorder the terms: -1.615384615 + 1.615384615 + -0.3846153846r + r2 = 0 + 1.615384615 Combine like terms: -1.615384615 + 1.615384615 = 0.000000000 0.000000000 + -0.3846153846r + r2 = 0 + 1.615384615 -0.3846153846r + r2 = 0 + 1.615384615 Combine like terms: 0 + 1.615384615 = 1.615384615 -0.3846153846r + r2 = 1.615384615 The r term is -0.3846153846r. Take half its coefficient (-0.1923076923). Square it (0.03698224852) and add it to both sides. Add '0.03698224852' to each side of the equation. -0.3846153846r + 0.03698224852 + r2 = 1.615384615 + 0.03698224852 Reorder the terms: 0.03698224852 + -0.3846153846r + r2 = 1.615384615 + 0.03698224852 Combine like terms: 1.615384615 + 0.03698224852 = 1.65236686352 0.03698224852 + -0.3846153846r + r2 = 1.65236686352 Factor a perfect square on the left side: (r + -0.1923076923)(r + -0.1923076923) = 1.65236686352 Calculate the square root of the right side: 1.285444228 Break this problem into two subproblems by setting (r + -0.1923076923) equal to 1.285444228 and -1.285444228.Subproblem 1
r + -0.1923076923 = 1.285444228 Simplifying r + -0.1923076923 = 1.285444228 Reorder the terms: -0.1923076923 + r = 1.285444228 Solving -0.1923076923 + r = 1.285444228 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.1923076923' to each side of the equation. -0.1923076923 + 0.1923076923 + r = 1.285444228 + 0.1923076923 Combine like terms: -0.1923076923 + 0.1923076923 = 0.0000000000 0.0000000000 + r = 1.285444228 + 0.1923076923 r = 1.285444228 + 0.1923076923 Combine like terms: 1.285444228 + 0.1923076923 = 1.4777519203 r = 1.4777519203 Simplifying r = 1.4777519203Subproblem 2
r + -0.1923076923 = -1.285444228 Simplifying r + -0.1923076923 = -1.285444228 Reorder the terms: -0.1923076923 + r = -1.285444228 Solving -0.1923076923 + r = -1.285444228 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.1923076923' to each side of the equation. -0.1923076923 + 0.1923076923 + r = -1.285444228 + 0.1923076923 Combine like terms: -0.1923076923 + 0.1923076923 = 0.0000000000 0.0000000000 + r = -1.285444228 + 0.1923076923 r = -1.285444228 + 0.1923076923 Combine like terms: -1.285444228 + 0.1923076923 = -1.0931365357 r = -1.0931365357 Simplifying r = -1.0931365357Solution
The solution to the problem is based on the solutions from the subproblems. r = {1.4777519203, -1.0931365357}
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