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Simplifying 7r2 = 13 + -14r Solving 7r2 = 13 + -14r Solving for variable 'r'. Reorder the terms: -13 + 14r + 7r2 = 13 + -14r + -13 + 14r Reorder the terms: -13 + 14r + 7r2 = 13 + -13 + -14r + 14r Combine like terms: 13 + -13 = 0 -13 + 14r + 7r2 = 0 + -14r + 14r -13 + 14r + 7r2 = -14r + 14r Combine like terms: -14r + 14r = 0 -13 + 14r + 7r2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -1.857142857 + 2r + r2 = 0 Move the constant term to the right: Add '1.857142857' to each side of the equation. -1.857142857 + 2r + 1.857142857 + r2 = 0 + 1.857142857 Reorder the terms: -1.857142857 + 1.857142857 + 2r + r2 = 0 + 1.857142857 Combine like terms: -1.857142857 + 1.857142857 = 0.000000000 0.000000000 + 2r + r2 = 0 + 1.857142857 2r + r2 = 0 + 1.857142857 Combine like terms: 0 + 1.857142857 = 1.857142857 2r + r2 = 1.857142857 The r term is 2r. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2r + 1 + r2 = 1.857142857 + 1 Reorder the terms: 1 + 2r + r2 = 1.857142857 + 1 Combine like terms: 1.857142857 + 1 = 2.857142857 1 + 2r + r2 = 2.857142857 Factor a perfect square on the left side: (r + 1)(r + 1) = 2.857142857 Calculate the square root of the right side: 1.690308509 Break this problem into two subproblems by setting (r + 1) equal to 1.690308509 and -1.690308509.Subproblem 1
r + 1 = 1.690308509 Simplifying r + 1 = 1.690308509 Reorder the terms: 1 + r = 1.690308509 Solving 1 + r = 1.690308509 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + r = 1.690308509 + -1 Combine like terms: 1 + -1 = 0 0 + r = 1.690308509 + -1 r = 1.690308509 + -1 Combine like terms: 1.690308509 + -1 = 0.690308509 r = 0.690308509 Simplifying r = 0.690308509Subproblem 2
r + 1 = -1.690308509 Simplifying r + 1 = -1.690308509 Reorder the terms: 1 + r = -1.690308509 Solving 1 + r = -1.690308509 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + r = -1.690308509 + -1 Combine like terms: 1 + -1 = 0 0 + r = -1.690308509 + -1 r = -1.690308509 + -1 Combine like terms: -1.690308509 + -1 = -2.690308509 r = -2.690308509 Simplifying r = -2.690308509Solution
The solution to the problem is based on the solutions from the subproblems. r = {0.690308509, -2.690308509}
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