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Simplifying 25k2 + 20k + 4 = 27 Reorder the terms: 4 + 20k + 25k2 = 27 Solving 4 + 20k + 25k2 = 27 Solving for variable 'k'. Reorder the terms: 4 + -27 + 20k + 25k2 = 27 + -27 Combine like terms: 4 + -27 = -23 -23 + 20k + 25k2 = 27 + -27 Combine like terms: 27 + -27 = 0 -23 + 20k + 25k2 = 0 Begin completing the square. Divide all terms by 25 the coefficient of the squared term: Divide each side by '25'. -0.92 + 0.8k + k2 = 0 Move the constant term to the right: Add '0.92' to each side of the equation. -0.92 + 0.8k + 0.92 + k2 = 0 + 0.92 Reorder the terms: -0.92 + 0.92 + 0.8k + k2 = 0 + 0.92 Combine like terms: -0.92 + 0.92 = 0.00 0.00 + 0.8k + k2 = 0 + 0.92 0.8k + k2 = 0 + 0.92 Combine like terms: 0 + 0.92 = 0.92 0.8k + k2 = 0.92 The k term is 0.8k. Take half its coefficient (0.4). Square it (0.16) and add it to both sides. Add '0.16' to each side of the equation. 0.8k + 0.16 + k2 = 0.92 + 0.16 Reorder the terms: 0.16 + 0.8k + k2 = 0.92 + 0.16 Combine like terms: 0.92 + 0.16 = 1.08 0.16 + 0.8k + k2 = 1.08 Factor a perfect square on the left side: (k + 0.4)(k + 0.4) = 1.08 Calculate the square root of the right side: 1.039230485 Break this problem into two subproblems by setting (k + 0.4) equal to 1.039230485 and -1.039230485.Subproblem 1
k + 0.4 = 1.039230485 Simplifying k + 0.4 = 1.039230485 Reorder the terms: 0.4 + k = 1.039230485 Solving 0.4 + k = 1.039230485 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.4' to each side of the equation. 0.4 + -0.4 + k = 1.039230485 + -0.4 Combine like terms: 0.4 + -0.4 = 0.0 0.0 + k = 1.039230485 + -0.4 k = 1.039230485 + -0.4 Combine like terms: 1.039230485 + -0.4 = 0.639230485 k = 0.639230485 Simplifying k = 0.639230485Subproblem 2
k + 0.4 = -1.039230485 Simplifying k + 0.4 = -1.039230485 Reorder the terms: 0.4 + k = -1.039230485 Solving 0.4 + k = -1.039230485 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.4' to each side of the equation. 0.4 + -0.4 + k = -1.039230485 + -0.4 Combine like terms: 0.4 + -0.4 = 0.0 0.0 + k = -1.039230485 + -0.4 k = -1.039230485 + -0.4 Combine like terms: -1.039230485 + -0.4 = -1.439230485 k = -1.439230485 Simplifying k = -1.439230485Solution
The solution to the problem is based on the solutions from the subproblems. k = {0.639230485, -1.439230485}
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