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Simplifying 0 = (3r2 + 7r + 1) + (4r2 + -8r + -2) Reorder the terms: 0 = (1 + 7r + 3r2) + (4r2 + -8r + -2) Remove parenthesis around (1 + 7r + 3r2) 0 = 1 + 7r + 3r2 + (4r2 + -8r + -2) Reorder the terms: 0 = 1 + 7r + 3r2 + (-2 + -8r + 4r2) Remove parenthesis around (-2 + -8r + 4r2) 0 = 1 + 7r + 3r2 + -2 + -8r + 4r2 Reorder the terms: 0 = 1 + -2 + 7r + -8r + 3r2 + 4r2 Combine like terms: 1 + -2 = -1 0 = -1 + 7r + -8r + 3r2 + 4r2 Combine like terms: 7r + -8r = -1r 0 = -1 + -1r + 3r2 + 4r2 Combine like terms: 3r2 + 4r2 = 7r2 0 = -1 + -1r + 7r2 Solving 0 = -1 + -1r + 7r2 Solving for variable 'r'. Combine like terms: 0 + 1 = 1 1 + r + -7r2 = -1 + -1r + 7r2 + 1 + r + -7r2 Reorder the terms: 1 + r + -7r2 = -1 + 1 + -1r + r + 7r2 + -7r2 Combine like terms: -1 + 1 = 0 1 + r + -7r2 = 0 + -1r + r + 7r2 + -7r2 1 + r + -7r2 = -1r + r + 7r2 + -7r2 Combine like terms: -1r + r = 0 1 + r + -7r2 = 0 + 7r2 + -7r2 1 + r + -7r2 = 7r2 + -7r2 Combine like terms: 7r2 + -7r2 = 0 1 + r + -7r2 = 0 Begin completing the square. Divide all terms by -7 the coefficient of the squared term: Divide each side by '-7'. -0.1428571429 + -0.1428571429r + r2 = 0 Move the constant term to the right: Add '0.1428571429' to each side of the equation. -0.1428571429 + -0.1428571429r + 0.1428571429 + r2 = 0 + 0.1428571429 Reorder the terms: -0.1428571429 + 0.1428571429 + -0.1428571429r + r2 = 0 + 0.1428571429 Combine like terms: -0.1428571429 + 0.1428571429 = 0.0000000000 0.0000000000 + -0.1428571429r + r2 = 0 + 0.1428571429 -0.1428571429r + r2 = 0 + 0.1428571429 Combine like terms: 0 + 0.1428571429 = 0.1428571429 -0.1428571429r + r2 = 0.1428571429 The r term is r. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -0.1428571429r + 0.25 + r2 = 0.1428571429 + 0.25 Reorder the terms: 0.25 + -0.1428571429r + r2 = 0.1428571429 + 0.25 Combine like terms: 0.1428571429 + 0.25 = 0.3928571429 0.25 + -0.1428571429r + r2 = 0.3928571429 Factor a perfect square on the left side: (r + 0.5)(r + 0.5) = 0.3928571429 Calculate the square root of the right side: 0.626783171 Break this problem into two subproblems by setting (r + 0.5) equal to 0.626783171 and -0.626783171.Subproblem 1
r + 0.5 = 0.626783171 Simplifying r + 0.5 = 0.626783171 Reorder the terms: 0.5 + r = 0.626783171 Solving 0.5 + r = 0.626783171 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + r = 0.626783171 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + r = 0.626783171 + -0.5 r = 0.626783171 + -0.5 Combine like terms: 0.626783171 + -0.5 = 0.126783171 r = 0.126783171 Simplifying r = 0.126783171Subproblem 2
r + 0.5 = -0.626783171 Simplifying r + 0.5 = -0.626783171 Reorder the terms: 0.5 + r = -0.626783171 Solving 0.5 + r = -0.626783171 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + r = -0.626783171 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + r = -0.626783171 + -0.5 r = -0.626783171 + -0.5 Combine like terms: -0.626783171 + -0.5 = -1.126783171 r = -1.126783171 Simplifying r = -1.126783171Solution
The solution to the problem is based on the solutions from the subproblems. r = {0.126783171, -1.126783171}
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