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Simplifying 10k2 = -3k + 23 Reorder the terms: 10k2 = 23 + -3k Solving 10k2 = 23 + -3k Solving for variable 'k'. Reorder the terms: -23 + 3k + 10k2 = 23 + -3k + -23 + 3k Reorder the terms: -23 + 3k + 10k2 = 23 + -23 + -3k + 3k Combine like terms: 23 + -23 = 0 -23 + 3k + 10k2 = 0 + -3k + 3k -23 + 3k + 10k2 = -3k + 3k Combine like terms: -3k + 3k = 0 -23 + 3k + 10k2 = 0 Begin completing the square. Divide all terms by 10 the coefficient of the squared term: Divide each side by '10'. -2.3 + 0.3k + k2 = 0 Move the constant term to the right: Add '2.3' to each side of the equation. -2.3 + 0.3k + 2.3 + k2 = 0 + 2.3 Reorder the terms: -2.3 + 2.3 + 0.3k + k2 = 0 + 2.3 Combine like terms: -2.3 + 2.3 = 0.0 0.0 + 0.3k + k2 = 0 + 2.3 0.3k + k2 = 0 + 2.3 Combine like terms: 0 + 2.3 = 2.3 0.3k + k2 = 2.3 The k term is 0.3k. Take half its coefficient (0.15). Square it (0.0225) and add it to both sides. Add '0.0225' to each side of the equation. 0.3k + 0.0225 + k2 = 2.3 + 0.0225 Reorder the terms: 0.0225 + 0.3k + k2 = 2.3 + 0.0225 Combine like terms: 2.3 + 0.0225 = 2.3225 0.0225 + 0.3k + k2 = 2.3225 Factor a perfect square on the left side: (k + 0.15)(k + 0.15) = 2.3225 Calculate the square root of the right side: 1.523975065 Break this problem into two subproblems by setting (k + 0.15) equal to 1.523975065 and -1.523975065.Subproblem 1
k + 0.15 = 1.523975065 Simplifying k + 0.15 = 1.523975065 Reorder the terms: 0.15 + k = 1.523975065 Solving 0.15 + k = 1.523975065 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.15' to each side of the equation. 0.15 + -0.15 + k = 1.523975065 + -0.15 Combine like terms: 0.15 + -0.15 = 0.00 0.00 + k = 1.523975065 + -0.15 k = 1.523975065 + -0.15 Combine like terms: 1.523975065 + -0.15 = 1.373975065 k = 1.373975065 Simplifying k = 1.373975065Subproblem 2
k + 0.15 = -1.523975065 Simplifying k + 0.15 = -1.523975065 Reorder the terms: 0.15 + k = -1.523975065 Solving 0.15 + k = -1.523975065 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.15' to each side of the equation. 0.15 + -0.15 + k = -1.523975065 + -0.15 Combine like terms: 0.15 + -0.15 = 0.00 0.00 + k = -1.523975065 + -0.15 k = -1.523975065 + -0.15 Combine like terms: -1.523975065 + -0.15 = -1.673975065 k = -1.673975065 Simplifying k = -1.673975065Solution
The solution to the problem is based on the solutions from the subproblems. k = {1.373975065, -1.673975065}
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