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Simplifying (r2 + 5) + -1(3r2 + -5r) = 0 Reorder the terms: (5 + r2) + -1(3r2 + -5r) = 0 Remove parenthesis around (5 + r2) 5 + r2 + -1(3r2 + -5r) = 0 Reorder the terms: 5 + r2 + -1(-5r + 3r2) = 0 5 + r2 + (-5r * -1 + 3r2 * -1) = 0 5 + r2 + (5r + -3r2) = 0 Reorder the terms: 5 + 5r + r2 + -3r2 = 0 Combine like terms: r2 + -3r2 = -2r2 5 + 5r + -2r2 = 0 Solving 5 + 5r + -2r2 = 0 Solving for variable 'r'. Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -2.5 + -2.5r + r2 = 0 Move the constant term to the right: Add '2.5' to each side of the equation. -2.5 + -2.5r + 2.5 + r2 = 0 + 2.5 Reorder the terms: -2.5 + 2.5 + -2.5r + r2 = 0 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + -2.5r + r2 = 0 + 2.5 -2.5r + r2 = 0 + 2.5 Combine like terms: 0 + 2.5 = 2.5 -2.5r + r2 = 2.5 The r term is -2.5r. Take half its coefficient (-1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. -2.5r + 1.5625 + r2 = 2.5 + 1.5625 Reorder the terms: 1.5625 + -2.5r + r2 = 2.5 + 1.5625 Combine like terms: 2.5 + 1.5625 = 4.0625 1.5625 + -2.5r + r2 = 4.0625 Factor a perfect square on the left side: (r + -1.25)(r + -1.25) = 4.0625 Calculate the square root of the right side: 2.015564437 Break this problem into two subproblems by setting (r + -1.25) equal to 2.015564437 and -2.015564437.Subproblem 1
r + -1.25 = 2.015564437 Simplifying r + -1.25 = 2.015564437 Reorder the terms: -1.25 + r = 2.015564437 Solving -1.25 + r = 2.015564437 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + r = 2.015564437 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + r = 2.015564437 + 1.25 r = 2.015564437 + 1.25 Combine like terms: 2.015564437 + 1.25 = 3.265564437 r = 3.265564437 Simplifying r = 3.265564437Subproblem 2
r + -1.25 = -2.015564437 Simplifying r + -1.25 = -2.015564437 Reorder the terms: -1.25 + r = -2.015564437 Solving -1.25 + r = -2.015564437 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + r = -2.015564437 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + r = -2.015564437 + 1.25 r = -2.015564437 + 1.25 Combine like terms: -2.015564437 + 1.25 = -0.765564437 r = -0.765564437 Simplifying r = -0.765564437Solution
The solution to the problem is based on the solutions from the subproblems. r = {3.265564437, -0.765564437}
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