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Simplifying -7k2 + -20k + 12 = 0 Reorder the terms: 12 + -20k + -7k2 = 0 Solving 12 + -20k + -7k2 = 0 Solving for variable 'k'. Begin completing the square. Divide all terms by -7 the coefficient of the squared term: Divide each side by '-7'. -1.714285714 + 2.857142857k + k2 = 0 Move the constant term to the right: Add '1.714285714' to each side of the equation. -1.714285714 + 2.857142857k + 1.714285714 + k2 = 0 + 1.714285714 Reorder the terms: -1.714285714 + 1.714285714 + 2.857142857k + k2 = 0 + 1.714285714 Combine like terms: -1.714285714 + 1.714285714 = 0.000000000 0.000000000 + 2.857142857k + k2 = 0 + 1.714285714 2.857142857k + k2 = 0 + 1.714285714 Combine like terms: 0 + 1.714285714 = 1.714285714 2.857142857k + k2 = 1.714285714 The k term is 2.857142857k. Take half its coefficient (1.428571429). Square it (2.040816328) and add it to both sides. Add '2.040816328' to each side of the equation. 2.857142857k + 2.040816328 + k2 = 1.714285714 + 2.040816328 Reorder the terms: 2.040816328 + 2.857142857k + k2 = 1.714285714 + 2.040816328 Combine like terms: 1.714285714 + 2.040816328 = 3.755102042 2.040816328 + 2.857142857k + k2 = 3.755102042 Factor a perfect square on the left side: (k + 1.428571429)(k + 1.428571429) = 3.755102042 Calculate the square root of the right side: 1.937808567 Break this problem into two subproblems by setting (k + 1.428571429) equal to 1.937808567 and -1.937808567.Subproblem 1
k + 1.428571429 = 1.937808567 Simplifying k + 1.428571429 = 1.937808567 Reorder the terms: 1.428571429 + k = 1.937808567 Solving 1.428571429 + k = 1.937808567 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-1.428571429' to each side of the equation. 1.428571429 + -1.428571429 + k = 1.937808567 + -1.428571429 Combine like terms: 1.428571429 + -1.428571429 = 0.000000000 0.000000000 + k = 1.937808567 + -1.428571429 k = 1.937808567 + -1.428571429 Combine like terms: 1.937808567 + -1.428571429 = 0.509237138 k = 0.509237138 Simplifying k = 0.509237138Subproblem 2
k + 1.428571429 = -1.937808567 Simplifying k + 1.428571429 = -1.937808567 Reorder the terms: 1.428571429 + k = -1.937808567 Solving 1.428571429 + k = -1.937808567 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-1.428571429' to each side of the equation. 1.428571429 + -1.428571429 + k = -1.937808567 + -1.428571429 Combine like terms: 1.428571429 + -1.428571429 = 0.000000000 0.000000000 + k = -1.937808567 + -1.428571429 k = -1.937808567 + -1.428571429 Combine like terms: -1.937808567 + -1.428571429 = -3.366379996 k = -3.366379996 Simplifying k = -3.366379996Solution
The solution to the problem is based on the solutions from the subproblems. k = {0.509237138, -3.366379996}
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