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Simplifying 21r2 + -41r + -42 = 0 Reorder the terms: -42 + -41r + 21r2 = 0 Solving -42 + -41r + 21r2 = 0 Solving for variable 'r'. Begin completing the square. Divide all terms by 21 the coefficient of the squared term: Divide each side by '21'. -2 + -1.952380952r + r2 = 0 Move the constant term to the right: Add '2' to each side of the equation. -2 + -1.952380952r + 2 + r2 = 0 + 2 Reorder the terms: -2 + 2 + -1.952380952r + r2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + -1.952380952r + r2 = 0 + 2 -1.952380952r + r2 = 0 + 2 Combine like terms: 0 + 2 = 2 -1.952380952r + r2 = 2 The r term is -1.952380952r. Take half its coefficient (-0.976190476). Square it (0.9529478454) and add it to both sides. Add '0.9529478454' to each side of the equation. -1.952380952r + 0.9529478454 + r2 = 2 + 0.9529478454 Reorder the terms: 0.9529478454 + -1.952380952r + r2 = 2 + 0.9529478454 Combine like terms: 2 + 0.9529478454 = 2.9529478454 0.9529478454 + -1.952380952r + r2 = 2.9529478454 Factor a perfect square on the left side: (r + -0.976190476)(r + -0.976190476) = 2.9529478454 Calculate the square root of the right side: 1.71841434 Break this problem into two subproblems by setting (r + -0.976190476) equal to 1.71841434 and -1.71841434.Subproblem 1
r + -0.976190476 = 1.71841434 Simplifying r + -0.976190476 = 1.71841434 Reorder the terms: -0.976190476 + r = 1.71841434 Solving -0.976190476 + r = 1.71841434 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.976190476' to each side of the equation. -0.976190476 + 0.976190476 + r = 1.71841434 + 0.976190476 Combine like terms: -0.976190476 + 0.976190476 = 0.000000000 0.000000000 + r = 1.71841434 + 0.976190476 r = 1.71841434 + 0.976190476 Combine like terms: 1.71841434 + 0.976190476 = 2.694604816 r = 2.694604816 Simplifying r = 2.694604816Subproblem 2
r + -0.976190476 = -1.71841434 Simplifying r + -0.976190476 = -1.71841434 Reorder the terms: -0.976190476 + r = -1.71841434 Solving -0.976190476 + r = -1.71841434 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.976190476' to each side of the equation. -0.976190476 + 0.976190476 + r = -1.71841434 + 0.976190476 Combine like terms: -0.976190476 + 0.976190476 = 0.000000000 0.000000000 + r = -1.71841434 + 0.976190476 r = -1.71841434 + 0.976190476 Combine like terms: -1.71841434 + 0.976190476 = -0.742223864 r = -0.742223864 Simplifying r = -0.742223864Solution
The solution to the problem is based on the solutions from the subproblems. r = {2.694604816, -0.742223864}
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