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Simplifying (19r2 + -3r + 3) + (14r + -7) = 0 Reorder the terms: (3 + -3r + 19r2) + (14r + -7) = 0 Remove parenthesis around (3 + -3r + 19r2) 3 + -3r + 19r2 + (14r + -7) = 0 Reorder the terms: 3 + -3r + 19r2 + (-7 + 14r) = 0 Remove parenthesis around (-7 + 14r) 3 + -3r + 19r2 + -7 + 14r = 0 Reorder the terms: 3 + -7 + -3r + 14r + 19r2 = 0 Combine like terms: 3 + -7 = -4 -4 + -3r + 14r + 19r2 = 0 Combine like terms: -3r + 14r = 11r -4 + 11r + 19r2 = 0 Solving -4 + 11r + 19r2 = 0 Solving for variable 'r'. Begin completing the square. Divide all terms by 19 the coefficient of the squared term: Divide each side by '19'. -0.2105263158 + 0.5789473684r + r2 = 0 Move the constant term to the right: Add '0.2105263158' to each side of the equation. -0.2105263158 + 0.5789473684r + 0.2105263158 + r2 = 0 + 0.2105263158 Reorder the terms: -0.2105263158 + 0.2105263158 + 0.5789473684r + r2 = 0 + 0.2105263158 Combine like terms: -0.2105263158 + 0.2105263158 = 0.0000000000 0.0000000000 + 0.5789473684r + r2 = 0 + 0.2105263158 0.5789473684r + r2 = 0 + 0.2105263158 Combine like terms: 0 + 0.2105263158 = 0.2105263158 0.5789473684r + r2 = 0.2105263158 The r term is 0.5789473684r. Take half its coefficient (0.2894736842). Square it (0.08379501384) and add it to both sides. Add '0.08379501384' to each side of the equation. 0.5789473684r + 0.08379501384 + r2 = 0.2105263158 + 0.08379501384 Reorder the terms: 0.08379501384 + 0.5789473684r + r2 = 0.2105263158 + 0.08379501384 Combine like terms: 0.2105263158 + 0.08379501384 = 0.29432132964 0.08379501384 + 0.5789473684r + r2 = 0.29432132964 Factor a perfect square on the left side: (r + 0.2894736842)(r + 0.2894736842) = 0.29432132964 Calculate the square root of the right side: 0.542513898 Break this problem into two subproblems by setting (r + 0.2894736842) equal to 0.542513898 and -0.542513898.Subproblem 1
r + 0.2894736842 = 0.542513898 Simplifying r + 0.2894736842 = 0.542513898 Reorder the terms: 0.2894736842 + r = 0.542513898 Solving 0.2894736842 + r = 0.542513898 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.2894736842' to each side of the equation. 0.2894736842 + -0.2894736842 + r = 0.542513898 + -0.2894736842 Combine like terms: 0.2894736842 + -0.2894736842 = 0.0000000000 0.0000000000 + r = 0.542513898 + -0.2894736842 r = 0.542513898 + -0.2894736842 Combine like terms: 0.542513898 + -0.2894736842 = 0.2530402138 r = 0.2530402138 Simplifying r = 0.2530402138Subproblem 2
r + 0.2894736842 = -0.542513898 Simplifying r + 0.2894736842 = -0.542513898 Reorder the terms: 0.2894736842 + r = -0.542513898 Solving 0.2894736842 + r = -0.542513898 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.2894736842' to each side of the equation. 0.2894736842 + -0.2894736842 + r = -0.542513898 + -0.2894736842 Combine like terms: 0.2894736842 + -0.2894736842 = 0.0000000000 0.0000000000 + r = -0.542513898 + -0.2894736842 r = -0.542513898 + -0.2894736842 Combine like terms: -0.542513898 + -0.2894736842 = -0.8319875822 r = -0.8319875822 Simplifying r = -0.8319875822Solution
The solution to the problem is based on the solutions from the subproblems. r = {0.2530402138, -0.8319875822}
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