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Simplifying k2 + -18k + 53 = 0 Reorder the terms: 53 + -18k + k2 = 0 Solving 53 + -18k + k2 = 0 Solving for variable 'k'. Begin completing the square. Move the constant term to the right: Add '-53' to each side of the equation. 53 + -18k + -53 + k2 = 0 + -53 Reorder the terms: 53 + -53 + -18k + k2 = 0 + -53 Combine like terms: 53 + -53 = 0 0 + -18k + k2 = 0 + -53 -18k + k2 = 0 + -53 Combine like terms: 0 + -53 = -53 -18k + k2 = -53 The k term is -18k. Take half its coefficient (-9). Square it (81) and add it to both sides. Add '81' to each side of the equation. -18k + 81 + k2 = -53 + 81 Reorder the terms: 81 + -18k + k2 = -53 + 81 Combine like terms: -53 + 81 = 28 81 + -18k + k2 = 28 Factor a perfect square on the left side: (k + -9)(k + -9) = 28 Calculate the square root of the right side: 5.291502622 Break this problem into two subproblems by setting (k + -9) equal to 5.291502622 and -5.291502622.Subproblem 1
k + -9 = 5.291502622 Simplifying k + -9 = 5.291502622 Reorder the terms: -9 + k = 5.291502622 Solving -9 + k = 5.291502622 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '9' to each side of the equation. -9 + 9 + k = 5.291502622 + 9 Combine like terms: -9 + 9 = 0 0 + k = 5.291502622 + 9 k = 5.291502622 + 9 Combine like terms: 5.291502622 + 9 = 14.291502622 k = 14.291502622 Simplifying k = 14.291502622Subproblem 2
k + -9 = -5.291502622 Simplifying k + -9 = -5.291502622 Reorder the terms: -9 + k = -5.291502622 Solving -9 + k = -5.291502622 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '9' to each side of the equation. -9 + 9 + k = -5.291502622 + 9 Combine like terms: -9 + 9 = 0 0 + k = -5.291502622 + 9 k = -5.291502622 + 9 Combine like terms: -5.291502622 + 9 = 3.708497378 k = 3.708497378 Simplifying k = 3.708497378Solution
The solution to the problem is based on the solutions from the subproblems. k = {14.291502622, 3.708497378}
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