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Simplifying 7k2 + 66k + 21 = 0 Reorder the terms: 21 + 66k + 7k2 = 0 Solving 21 + 66k + 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'. 3 + 9.428571429k + k2 = 0 Move the constant term to the right: Add '-3' to each side of the equation. 3 + 9.428571429k + -3 + k2 = 0 + -3 Reorder the terms: 3 + -3 + 9.428571429k + k2 = 0 + -3 Combine like terms: 3 + -3 = 0 0 + 9.428571429k + k2 = 0 + -3 9.428571429k + k2 = 0 + -3 Combine like terms: 0 + -3 = -3 9.428571429k + k2 = -3 The k term is 9.428571429k. Take half its coefficient (4.714285715). Square it (22.22448980) and add it to both sides. Add '22.22448980' to each side of the equation. 9.428571429k + 22.22448980 + k2 = -3 + 22.22448980 Reorder the terms: 22.22448980 + 9.428571429k + k2 = -3 + 22.22448980 Combine like terms: -3 + 22.22448980 = 19.2244898 22.22448980 + 9.428571429k + k2 = 19.2244898 Factor a perfect square on the left side: (k + 4.714285715)(k + 4.714285715) = 19.2244898 Calculate the square root of the right side: 4.384574073 Break this problem into two subproblems by setting (k + 4.714285715) equal to 4.384574073 and -4.384574073.Subproblem 1
k + 4.714285715 = 4.384574073 Simplifying k + 4.714285715 = 4.384574073 Reorder the terms: 4.714285715 + k = 4.384574073 Solving 4.714285715 + k = 4.384574073 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-4.714285715' to each side of the equation. 4.714285715 + -4.714285715 + k = 4.384574073 + -4.714285715 Combine like terms: 4.714285715 + -4.714285715 = 0.000000000 0.000000000 + k = 4.384574073 + -4.714285715 k = 4.384574073 + -4.714285715 Combine like terms: 4.384574073 + -4.714285715 = -0.329711642 k = -0.329711642 Simplifying k = -0.329711642Subproblem 2
k + 4.714285715 = -4.384574073 Simplifying k + 4.714285715 = -4.384574073 Reorder the terms: 4.714285715 + k = -4.384574073 Solving 4.714285715 + k = -4.384574073 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-4.714285715' to each side of the equation. 4.714285715 + -4.714285715 + k = -4.384574073 + -4.714285715 Combine like terms: 4.714285715 + -4.714285715 = 0.000000000 0.000000000 + k = -4.384574073 + -4.714285715 k = -4.384574073 + -4.714285715 Combine like terms: -4.384574073 + -4.714285715 = -9.098859788 k = -9.098859788 Simplifying k = -9.098859788Solution
The solution to the problem is based on the solutions from the subproblems. k = {-0.329711642, -9.098859788}
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