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Simplifying 20k = 18 + k2 Solving 20k = 18 + k2 Solving for variable 'k'. Reorder the terms: -18 + 20k + -1k2 = 18 + k2 + -18 + -1k2 Reorder the terms: -18 + 20k + -1k2 = 18 + -18 + k2 + -1k2 Combine like terms: 18 + -18 = 0 -18 + 20k + -1k2 = 0 + k2 + -1k2 -18 + 20k + -1k2 = k2 + -1k2 Combine like terms: k2 + -1k2 = 0 -18 + 20k + -1k2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 18 + -20k + k2 = 0 Move the constant term to the right: Add '-18' to each side of the equation. 18 + -20k + -18 + k2 = 0 + -18 Reorder the terms: 18 + -18 + -20k + k2 = 0 + -18 Combine like terms: 18 + -18 = 0 0 + -20k + k2 = 0 + -18 -20k + k2 = 0 + -18 Combine like terms: 0 + -18 = -18 -20k + k2 = -18 The k term is -20k. Take half its coefficient (-10). Square it (100) and add it to both sides. Add '100' to each side of the equation. -20k + 100 + k2 = -18 + 100 Reorder the terms: 100 + -20k + k2 = -18 + 100 Combine like terms: -18 + 100 = 82 100 + -20k + k2 = 82 Factor a perfect square on the left side: (k + -10)(k + -10) = 82 Calculate the square root of the right side: 9.055385138 Break this problem into two subproblems by setting (k + -10) equal to 9.055385138 and -9.055385138.Subproblem 1
k + -10 = 9.055385138 Simplifying k + -10 = 9.055385138 Reorder the terms: -10 + k = 9.055385138 Solving -10 + k = 9.055385138 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '10' to each side of the equation. -10 + 10 + k = 9.055385138 + 10 Combine like terms: -10 + 10 = 0 0 + k = 9.055385138 + 10 k = 9.055385138 + 10 Combine like terms: 9.055385138 + 10 = 19.055385138 k = 19.055385138 Simplifying k = 19.055385138Subproblem 2
k + -10 = -9.055385138 Simplifying k + -10 = -9.055385138 Reorder the terms: -10 + k = -9.055385138 Solving -10 + k = -9.055385138 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '10' to each side of the equation. -10 + 10 + k = -9.055385138 + 10 Combine like terms: -10 + 10 = 0 0 + k = -9.055385138 + 10 k = -9.055385138 + 10 Combine like terms: -9.055385138 + 10 = 0.944614862 k = 0.944614862 Simplifying k = 0.944614862Solution
The solution to the problem is based on the solutions from the subproblems. k = {19.055385138, 0.944614862}
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