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Simplifying 2k2 + -20 = -1k Reorder the terms: -20 + 2k2 = -1k Solving -20 + 2k2 = -1k Solving for variable 'k'. Reorder the terms: -20 + k + 2k2 = -1k + k Combine like terms: -1k + k = 0 -20 + k + 2k2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -10 + 0.5k + k2 = 0 Move the constant term to the right: Add '10' to each side of the equation. -10 + 0.5k + 10 + k2 = 0 + 10 Reorder the terms: -10 + 10 + 0.5k + k2 = 0 + 10 Combine like terms: -10 + 10 = 0 0 + 0.5k + k2 = 0 + 10 0.5k + k2 = 0 + 10 Combine like terms: 0 + 10 = 10 0.5k + k2 = 10 The k term is k. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. 0.5k + 0.25 + k2 = 10 + 0.25 Reorder the terms: 0.25 + 0.5k + k2 = 10 + 0.25 Combine like terms: 10 + 0.25 = 10.25 0.25 + 0.5k + k2 = 10.25 Factor a perfect square on the left side: (k + 0.5)(k + 0.5) = 10.25 Calculate the square root of the right side: 3.201562119 Break this problem into two subproblems by setting (k + 0.5) equal to 3.201562119 and -3.201562119.Subproblem 1
k + 0.5 = 3.201562119 Simplifying k + 0.5 = 3.201562119 Reorder the terms: 0.5 + k = 3.201562119 Solving 0.5 + k = 3.201562119 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + k = 3.201562119 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + k = 3.201562119 + -0.5 k = 3.201562119 + -0.5 Combine like terms: 3.201562119 + -0.5 = 2.701562119 k = 2.701562119 Simplifying k = 2.701562119Subproblem 2
k + 0.5 = -3.201562119 Simplifying k + 0.5 = -3.201562119 Reorder the terms: 0.5 + k = -3.201562119 Solving 0.5 + k = -3.201562119 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + k = -3.201562119 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + k = -3.201562119 + -0.5 k = -3.201562119 + -0.5 Combine like terms: -3.201562119 + -0.5 = -3.701562119 k = -3.701562119 Simplifying k = -3.701562119Solution
The solution to the problem is based on the solutions from the subproblems. k = {2.701562119, -3.701562119}
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