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Simplifying 3k2 + 11 = 18k Reorder the terms: 11 + 3k2 = 18k Solving 11 + 3k2 = 18k Solving for variable 'k'. Reorder the terms: 11 + -18k + 3k2 = 18k + -18k Combine like terms: 18k + -18k = 0 11 + -18k + 3k2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 3.666666667 + -6k + k2 = 0 Move the constant term to the right: Add '-3.666666667' to each side of the equation. 3.666666667 + -6k + -3.666666667 + k2 = 0 + -3.666666667 Reorder the terms: 3.666666667 + -3.666666667 + -6k + k2 = 0 + -3.666666667 Combine like terms: 3.666666667 + -3.666666667 = 0.000000000 0.000000000 + -6k + k2 = 0 + -3.666666667 -6k + k2 = 0 + -3.666666667 Combine like terms: 0 + -3.666666667 = -3.666666667 -6k + k2 = -3.666666667 The k term is -6k. Take half its coefficient (-3). Square it (9) and add it to both sides. Add '9' to each side of the equation. -6k + 9 + k2 = -3.666666667 + 9 Reorder the terms: 9 + -6k + k2 = -3.666666667 + 9 Combine like terms: -3.666666667 + 9 = 5.333333333 9 + -6k + k2 = 5.333333333 Factor a perfect square on the left side: (k + -3)(k + -3) = 5.333333333 Calculate the square root of the right side: 2.309401077 Break this problem into two subproblems by setting (k + -3) equal to 2.309401077 and -2.309401077.Subproblem 1
k + -3 = 2.309401077 Simplifying k + -3 = 2.309401077 Reorder the terms: -3 + k = 2.309401077 Solving -3 + k = 2.309401077 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + k = 2.309401077 + 3 Combine like terms: -3 + 3 = 0 0 + k = 2.309401077 + 3 k = 2.309401077 + 3 Combine like terms: 2.309401077 + 3 = 5.309401077 k = 5.309401077 Simplifying k = 5.309401077Subproblem 2
k + -3 = -2.309401077 Simplifying k + -3 = -2.309401077 Reorder the terms: -3 + k = -2.309401077 Solving -3 + k = -2.309401077 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + k = -2.309401077 + 3 Combine like terms: -3 + 3 = 0 0 + k = -2.309401077 + 3 k = -2.309401077 + 3 Combine like terms: -2.309401077 + 3 = 0.690598923 k = 0.690598923 Simplifying k = 0.690598923Solution
The solution to the problem is based on the solutions from the subproblems. k = {5.309401077, 0.690598923}
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