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Simplifying (3r2 + 7r + 1) + (4r2 + -8 + -2) = 0 Reorder the terms: (1 + 7r + 3r2) + (4r2 + -8 + -2) = 0 Remove parenthesis around (1 + 7r + 3r2) 1 + 7r + 3r2 + (4r2 + -8 + -2) = 0 Reorder the terms: 1 + 7r + 3r2 + (-8 + -2 + 4r2) = 0 Combine like terms: -8 + -2 = -10 1 + 7r + 3r2 + (-10 + 4r2) = 0 Remove parenthesis around (-10 + 4r2) 1 + 7r + 3r2 + -10 + 4r2 = 0 Reorder the terms: 1 + -10 + 7r + 3r2 + 4r2 = 0 Combine like terms: 1 + -10 = -9 -9 + 7r + 3r2 + 4r2 = 0 Combine like terms: 3r2 + 4r2 = 7r2 -9 + 7r + 7r2 = 0 Solving -9 + 7r + 7r2 = 0 Solving for variable 'r'. Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -1.285714286 + r + r2 = 0 Move the constant term to the right: Add '1.285714286' to each side of the equation. -1.285714286 + r + 1.285714286 + r2 = 0 + 1.285714286 Reorder the terms: -1.285714286 + 1.285714286 + r + r2 = 0 + 1.285714286 Combine like terms: -1.285714286 + 1.285714286 = 0.000000000 0.000000000 + r + r2 = 0 + 1.285714286 r + r2 = 0 + 1.285714286 Combine like terms: 0 + 1.285714286 = 1.285714286 r + r2 = 1.285714286 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. + 0.25 + r2 = 1.285714286 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + r2 = 1.285714286 + 0.25 Combine like terms: 1.285714286 + 0.25 = 1.535714286 1.25 + r2 = 1.535714286 Factor a perfect square on the left side: (r + 0.5)(r + 0.5) = 1.535714286 Calculate the square root of the right side: 1.239239398 Break this problem into two subproblems by setting (r + 0.5) equal to 1.239239398 and -1.239239398.Subproblem 1
r + 0.5 = 1.239239398 Simplifying r + 0.5 = 1.239239398 Reorder the terms: 0.5 + r = 1.239239398 Solving 0.5 + r = 1.239239398 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 = 1.239239398 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + r = 1.239239398 + -0.5 r = 1.239239398 + -0.5 Combine like terms: 1.239239398 + -0.5 = 0.739239398 r = 0.739239398 Simplifying r = 0.739239398Subproblem 2
r + 0.5 = -1.239239398 Simplifying r + 0.5 = -1.239239398 Reorder the terms: 0.5 + r = -1.239239398 Solving 0.5 + r = -1.239239398 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 = -1.239239398 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + r = -1.239239398 + -0.5 r = -1.239239398 + -0.5 Combine like terms: -1.239239398 + -0.5 = -1.739239398 r = -1.739239398 Simplifying r = -1.739239398Solution
The solution to the problem is based on the solutions from the subproblems. r = {0.739239398, -1.739239398}
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