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Simplifying 8r * 8r = 6 + -13r Reorder the terms for easier multiplication: 8 * 8r * r = 6 + -13r Multiply 8 * 8 64r * r = 6 + -13r Multiply r * r 64r2 = 6 + -13r Solving 64r2 = 6 + -13r Solving for variable 'r'. Reorder the terms: -6 + 13r + 64r2 = 6 + -13r + -6 + 13r Reorder the terms: -6 + 13r + 64r2 = 6 + -6 + -13r + 13r Combine like terms: 6 + -6 = 0 -6 + 13r + 64r2 = 0 + -13r + 13r -6 + 13r + 64r2 = -13r + 13r Combine like terms: -13r + 13r = 0 -6 + 13r + 64r2 = 0 Begin completing the square. Divide all terms by 64 the coefficient of the squared term: Divide each side by '64'. -0.09375 + 0.203125r + r2 = 0 Move the constant term to the right: Add '0.09375' to each side of the equation. -0.09375 + 0.203125r + 0.09375 + r2 = 0 + 0.09375 Reorder the terms: -0.09375 + 0.09375 + 0.203125r + r2 = 0 + 0.09375 Combine like terms: -0.09375 + 0.09375 = 0.00000 0.00000 + 0.203125r + r2 = 0 + 0.09375 0.203125r + r2 = 0 + 0.09375 Combine like terms: 0 + 0.09375 = 0.09375 0.203125r + r2 = 0.09375 The r term is 0.203125r. Take half its coefficient (0.1015625). Square it (0.01031494141) and add it to both sides. Add '0.01031494141' to each side of the equation. 0.203125r + 0.01031494141 + r2 = 0.09375 + 0.01031494141 Reorder the terms: 0.01031494141 + 0.203125r + r2 = 0.09375 + 0.01031494141 Combine like terms: 0.09375 + 0.01031494141 = 0.10406494141 0.01031494141 + 0.203125r + r2 = 0.10406494141 Factor a perfect square on the left side: (r + 0.1015625)(r + 0.1015625) = 0.10406494141 Calculate the square root of the right side: 0.322590982 Break this problem into two subproblems by setting (r + 0.1015625) equal to 0.322590982 and -0.322590982.Subproblem 1
r + 0.1015625 = 0.322590982 Simplifying r + 0.1015625 = 0.322590982 Reorder the terms: 0.1015625 + r = 0.322590982 Solving 0.1015625 + r = 0.322590982 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.1015625' to each side of the equation. 0.1015625 + -0.1015625 + r = 0.322590982 + -0.1015625 Combine like terms: 0.1015625 + -0.1015625 = 0.0000000 0.0000000 + r = 0.322590982 + -0.1015625 r = 0.322590982 + -0.1015625 Combine like terms: 0.322590982 + -0.1015625 = 0.221028482 r = 0.221028482 Simplifying r = 0.221028482Subproblem 2
r + 0.1015625 = -0.322590982 Simplifying r + 0.1015625 = -0.322590982 Reorder the terms: 0.1015625 + r = -0.322590982 Solving 0.1015625 + r = -0.322590982 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.1015625' to each side of the equation. 0.1015625 + -0.1015625 + r = -0.322590982 + -0.1015625 Combine like terms: 0.1015625 + -0.1015625 = 0.0000000 0.0000000 + r = -0.322590982 + -0.1015625 r = -0.322590982 + -0.1015625 Combine like terms: -0.322590982 + -0.1015625 = -0.424153482 r = -0.424153482 Simplifying r = -0.424153482Solution
The solution to the problem is based on the solutions from the subproblems. r = {0.221028482, -0.424153482}
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