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Simplifying 7k2 + -12k + -12 = 0 Reorder the terms: -12 + -12k + 7k2 = 0 Solving -12 + -12k + 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 + -1.714285714k + k2 = 0 Move the constant term to the right: Add '1.714285714' to each side of the equation. -1.714285714 + -1.714285714k + 1.714285714 + k2 = 0 + 1.714285714 Reorder the terms: -1.714285714 + 1.714285714 + -1.714285714k + k2 = 0 + 1.714285714 Combine like terms: -1.714285714 + 1.714285714 = 0.000000000 0.000000000 + -1.714285714k + k2 = 0 + 1.714285714 -1.714285714k + k2 = 0 + 1.714285714 Combine like terms: 0 + 1.714285714 = 1.714285714 -1.714285714k + k2 = 1.714285714 The k term is -1.714285714k. Take half its coefficient (-0.857142857). Square it (0.7346938773) and add it to both sides. Add '0.7346938773' to each side of the equation. -1.714285714k + 0.7346938773 + k2 = 1.714285714 + 0.7346938773 Reorder the terms: 0.7346938773 + -1.714285714k + k2 = 1.714285714 + 0.7346938773 Combine like terms: 1.714285714 + 0.7346938773 = 2.4489795913 0.7346938773 + -1.714285714k + k2 = 2.4489795913 Factor a perfect square on the left side: (k + -0.857142857)(k + -0.857142857) = 2.4489795913 Calculate the square root of the right side: 1.564921593 Break this problem into two subproblems by setting (k + -0.857142857) equal to 1.564921593 and -1.564921593.Subproblem 1
k + -0.857142857 = 1.564921593 Simplifying k + -0.857142857 = 1.564921593 Reorder the terms: -0.857142857 + k = 1.564921593 Solving -0.857142857 + k = 1.564921593 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.857142857' to each side of the equation. -0.857142857 + 0.857142857 + k = 1.564921593 + 0.857142857 Combine like terms: -0.857142857 + 0.857142857 = 0.000000000 0.000000000 + k = 1.564921593 + 0.857142857 k = 1.564921593 + 0.857142857 Combine like terms: 1.564921593 + 0.857142857 = 2.42206445 k = 2.42206445 Simplifying k = 2.42206445Subproblem 2
k + -0.857142857 = -1.564921593 Simplifying k + -0.857142857 = -1.564921593 Reorder the terms: -0.857142857 + k = -1.564921593 Solving -0.857142857 + k = -1.564921593 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.857142857' to each side of the equation. -0.857142857 + 0.857142857 + k = -1.564921593 + 0.857142857 Combine like terms: -0.857142857 + 0.857142857 = 0.000000000 0.000000000 + k = -1.564921593 + 0.857142857 k = -1.564921593 + 0.857142857 Combine like terms: -1.564921593 + 0.857142857 = -0.707778736 k = -0.707778736 Simplifying k = -0.707778736Solution
The solution to the problem is based on the solutions from the subproblems. k = {2.42206445, -0.707778736}
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