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Simplifying 7k2 = 15 + 10k Solving 7k2 = 15 + 10k Solving for variable 'k'. Reorder the terms: -15 + -10k + 7k2 = 15 + 10k + -15 + -10k Reorder the terms: -15 + -10k + 7k2 = 15 + -15 + 10k + -10k Combine like terms: 15 + -15 = 0 -15 + -10k + 7k2 = 0 + 10k + -10k -15 + -10k + 7k2 = 10k + -10k Combine like terms: 10k + -10k = 0 -15 + -10k + 7k2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -2.142857143 + -1.428571429k + k2 = 0 Move the constant term to the right: Add '2.142857143' to each side of the equation. -2.142857143 + -1.428571429k + 2.142857143 + k2 = 0 + 2.142857143 Reorder the terms: -2.142857143 + 2.142857143 + -1.428571429k + k2 = 0 + 2.142857143 Combine like terms: -2.142857143 + 2.142857143 = 0.000000000 0.000000000 + -1.428571429k + k2 = 0 + 2.142857143 -1.428571429k + k2 = 0 + 2.142857143 Combine like terms: 0 + 2.142857143 = 2.142857143 -1.428571429k + k2 = 2.142857143 The k term is -1.428571429k. Take half its coefficient (-0.7142857145). Square it (0.5102040819) and add it to both sides. Add '0.5102040819' to each side of the equation. -1.428571429k + 0.5102040819 + k2 = 2.142857143 + 0.5102040819 Reorder the terms: 0.5102040819 + -1.428571429k + k2 = 2.142857143 + 0.5102040819 Combine like terms: 2.142857143 + 0.5102040819 = 2.6530612249 0.5102040819 + -1.428571429k + k2 = 2.6530612249 Factor a perfect square on the left side: (k + -0.7142857145)(k + -0.7142857145) = 2.6530612249 Calculate the square root of the right side: 1.628822036 Break this problem into two subproblems by setting (k + -0.7142857145) equal to 1.628822036 and -1.628822036.Subproblem 1
k + -0.7142857145 = 1.628822036 Simplifying k + -0.7142857145 = 1.628822036 Reorder the terms: -0.7142857145 + k = 1.628822036 Solving -0.7142857145 + k = 1.628822036 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.7142857145' to each side of the equation. -0.7142857145 + 0.7142857145 + k = 1.628822036 + 0.7142857145 Combine like terms: -0.7142857145 + 0.7142857145 = 0.0000000000 0.0000000000 + k = 1.628822036 + 0.7142857145 k = 1.628822036 + 0.7142857145 Combine like terms: 1.628822036 + 0.7142857145 = 2.3431077505 k = 2.3431077505 Simplifying k = 2.3431077505Subproblem 2
k + -0.7142857145 = -1.628822036 Simplifying k + -0.7142857145 = -1.628822036 Reorder the terms: -0.7142857145 + k = -1.628822036 Solving -0.7142857145 + k = -1.628822036 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.7142857145' to each side of the equation. -0.7142857145 + 0.7142857145 + k = -1.628822036 + 0.7142857145 Combine like terms: -0.7142857145 + 0.7142857145 = 0.0000000000 0.0000000000 + k = -1.628822036 + 0.7142857145 k = -1.628822036 + 0.7142857145 Combine like terms: -1.628822036 + 0.7142857145 = -0.9145363215 k = -0.9145363215 Simplifying k = -0.9145363215Solution
The solution to the problem is based on the solutions from the subproblems. k = {2.3431077505, -0.9145363215}
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