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Simplifying 3k2 + 15k + 14 = 0 Reorder the terms: 14 + 15k + 3k2 = 0 Solving 14 + 15k + 3k2 = 0 Solving for variable 'k'. Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 4.666666667 + 5k + k2 = 0 Move the constant term to the right: Add '-4.666666667' to each side of the equation. 4.666666667 + 5k + -4.666666667 + k2 = 0 + -4.666666667 Reorder the terms: 4.666666667 + -4.666666667 + 5k + k2 = 0 + -4.666666667 Combine like terms: 4.666666667 + -4.666666667 = 0.000000000 0.000000000 + 5k + k2 = 0 + -4.666666667 5k + k2 = 0 + -4.666666667 Combine like terms: 0 + -4.666666667 = -4.666666667 5k + k2 = -4.666666667 The k term is 5k. Take half its coefficient (2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. 5k + 6.25 + k2 = -4.666666667 + 6.25 Reorder the terms: 6.25 + 5k + k2 = -4.666666667 + 6.25 Combine like terms: -4.666666667 + 6.25 = 1.583333333 6.25 + 5k + k2 = 1.583333333 Factor a perfect square on the left side: (k + 2.5)(k + 2.5) = 1.583333333 Calculate the square root of the right side: 1.258305739 Break this problem into two subproblems by setting (k + 2.5) equal to 1.258305739 and -1.258305739.Subproblem 1
k + 2.5 = 1.258305739 Simplifying k + 2.5 = 1.258305739 Reorder the terms: 2.5 + k = 1.258305739 Solving 2.5 + k = 1.258305739 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + k = 1.258305739 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + k = 1.258305739 + -2.5 k = 1.258305739 + -2.5 Combine like terms: 1.258305739 + -2.5 = -1.241694261 k = -1.241694261 Simplifying k = -1.241694261Subproblem 2
k + 2.5 = -1.258305739 Simplifying k + 2.5 = -1.258305739 Reorder the terms: 2.5 + k = -1.258305739 Solving 2.5 + k = -1.258305739 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + k = -1.258305739 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + k = -1.258305739 + -2.5 k = -1.258305739 + -2.5 Combine like terms: -1.258305739 + -2.5 = -3.758305739 k = -3.758305739 Simplifying k = -3.758305739Solution
The solution to the problem is based on the solutions from the subproblems. k = {-1.241694261, -3.758305739}
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