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Simplifying 5(3r + -4) + -1(r2 + 2) + -2r(r + -3) = 0 Reorder the terms: 5(-4 + 3r) + -1(r2 + 2) + -2r(r + -3) = 0 (-4 * 5 + 3r * 5) + -1(r2 + 2) + -2r(r + -3) = 0 (-20 + 15r) + -1(r2 + 2) + -2r(r + -3) = 0 Reorder the terms: -20 + 15r + -1(2 + r2) + -2r(r + -3) = 0 -20 + 15r + (2 * -1 + r2 * -1) + -2r(r + -3) = 0 -20 + 15r + (-2 + -1r2) + -2r(r + -3) = 0 Reorder the terms: -20 + 15r + -2 + -1r2 + -2r(-3 + r) = 0 -20 + 15r + -2 + -1r2 + (-3 * -2r + r * -2r) = 0 -20 + 15r + -2 + -1r2 + (6r + -2r2) = 0 Reorder the terms: -20 + -2 + 15r + 6r + -1r2 + -2r2 = 0 Combine like terms: -20 + -2 = -22 -22 + 15r + 6r + -1r2 + -2r2 = 0 Combine like terms: 15r + 6r = 21r -22 + 21r + -1r2 + -2r2 = 0 Combine like terms: -1r2 + -2r2 = -3r2 -22 + 21r + -3r2 = 0 Solving -22 + 21r + -3r2 = 0 Solving for variable 'r'. Begin completing the square. Divide all terms by -3 the coefficient of the squared term: Divide each side by '-3'. 7.333333333 + -7r + r2 = 0 Move the constant term to the right: Add '-7.333333333' to each side of the equation. 7.333333333 + -7r + -7.333333333 + r2 = 0 + -7.333333333 Reorder the terms: 7.333333333 + -7.333333333 + -7r + r2 = 0 + -7.333333333 Combine like terms: 7.333333333 + -7.333333333 = 0.000000000 0.000000000 + -7r + r2 = 0 + -7.333333333 -7r + r2 = 0 + -7.333333333 Combine like terms: 0 + -7.333333333 = -7.333333333 -7r + r2 = -7.333333333 The r term is -7r. Take half its coefficient (-3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. -7r + 12.25 + r2 = -7.333333333 + 12.25 Reorder the terms: 12.25 + -7r + r2 = -7.333333333 + 12.25 Combine like terms: -7.333333333 + 12.25 = 4.916666667 12.25 + -7r + r2 = 4.916666667 Factor a perfect square on the left side: (r + -3.5)(r + -3.5) = 4.916666667 Calculate the square root of the right side: 2.217355783 Break this problem into two subproblems by setting (r + -3.5) equal to 2.217355783 and -2.217355783.Subproblem 1
r + -3.5 = 2.217355783 Simplifying r + -3.5 = 2.217355783 Reorder the terms: -3.5 + r = 2.217355783 Solving -3.5 + r = 2.217355783 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + r = 2.217355783 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + r = 2.217355783 + 3.5 r = 2.217355783 + 3.5 Combine like terms: 2.217355783 + 3.5 = 5.717355783 r = 5.717355783 Simplifying r = 5.717355783Subproblem 2
r + -3.5 = -2.217355783 Simplifying r + -3.5 = -2.217355783 Reorder the terms: -3.5 + r = -2.217355783 Solving -3.5 + r = -2.217355783 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + r = -2.217355783 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + r = -2.217355783 + 3.5 r = -2.217355783 + 3.5 Combine like terms: -2.217355783 + 3.5 = 1.282644217 r = 1.282644217 Simplifying r = 1.282644217Solution
The solution to the problem is based on the solutions from the subproblems. r = {5.717355783, 1.282644217}
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