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Simplifying 14093.89 = 628r2 + 617.952r Reorder the terms: 14093.89 = 617.952r + 628r2 Solving 14093.89 = 617.952r + 628r2 Solving for variable 'r'. Reorder the terms: 14093.89 + -617.952r + -628r2 = 617.952r + -617.952r + 628r2 + -628r2 Combine like terms: 617.952r + -617.952r = 0.000 14093.89 + -617.952r + -628r2 = 0.000 + 628r2 + -628r2 14093.89 + -617.952r + -628r2 = 628r2 + -628r2 Combine like terms: 628r2 + -628r2 = 0 14093.89 + -617.952r + -628r2 = 0 Begin completing the square. Divide all terms by -628 the coefficient of the squared term: Divide each side by '-628'. -22.4425 + 0.984r + r2 = 0 Move the constant term to the right: Add '22.4425' to each side of the equation. -22.4425 + 0.984r + 22.4425 + r2 = 0 + 22.4425 Reorder the terms: -22.4425 + 22.4425 + 0.984r + r2 = 0 + 22.4425 Combine like terms: -22.4425 + 22.4425 = 0.0000 0.0000 + 0.984r + r2 = 0 + 22.4425 0.984r + r2 = 0 + 22.4425 Combine like terms: 0 + 22.4425 = 22.4425 0.984r + r2 = 22.4425 The r term is 0.984r. Take half its coefficient (0.492). Square it (0.242064) and add it to both sides. Add '0.242064' to each side of the equation. 0.984r + 0.242064 + r2 = 22.4425 + 0.242064 Reorder the terms: 0.242064 + 0.984r + r2 = 22.4425 + 0.242064 Combine like terms: 22.4425 + 0.242064 = 22.684564 0.242064 + 0.984r + r2 = 22.684564 Factor a perfect square on the left side: (r + 0.492)(r + 0.492) = 22.684564 Calculate the square root of the right side: 4.762831511 Break this problem into two subproblems by setting (r + 0.492) equal to 4.762831511 and -4.762831511.Subproblem 1
r + 0.492 = 4.762831511 Simplifying r + 0.492 = 4.762831511 Reorder the terms: 0.492 + r = 4.762831511 Solving 0.492 + r = 4.762831511 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.492' to each side of the equation. 0.492 + -0.492 + r = 4.762831511 + -0.492 Combine like terms: 0.492 + -0.492 = 0.000 0.000 + r = 4.762831511 + -0.492 r = 4.762831511 + -0.492 Combine like terms: 4.762831511 + -0.492 = 4.270831511 r = 4.270831511 Simplifying r = 4.270831511Subproblem 2
r + 0.492 = -4.762831511 Simplifying r + 0.492 = -4.762831511 Reorder the terms: 0.492 + r = -4.762831511 Solving 0.492 + r = -4.762831511 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.492' to each side of the equation. 0.492 + -0.492 + r = -4.762831511 + -0.492 Combine like terms: 0.492 + -0.492 = 0.000 0.000 + r = -4.762831511 + -0.492 r = -4.762831511 + -0.492 Combine like terms: -4.762831511 + -0.492 = -5.254831511 r = -5.254831511 Simplifying r = -5.254831511Solution
The solution to the problem is based on the solutions from the subproblems. r = {4.270831511, -5.254831511}
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