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Simplifying 3k + -15 + k * k + 4k = 0 Multiply k * k 3k + -15 + k2 + 4k = 0 Reorder the terms: -15 + 3k + 4k + k2 = 0 Combine like terms: 3k + 4k = 7k -15 + 7k + k2 = 0 Solving -15 + 7k + k2 = 0 Solving for variable 'k'. Begin completing the square. Move the constant term to the right: Add '15' to each side of the equation. -15 + 7k + 15 + k2 = 0 + 15 Reorder the terms: -15 + 15 + 7k + k2 = 0 + 15 Combine like terms: -15 + 15 = 0 0 + 7k + k2 = 0 + 15 7k + k2 = 0 + 15 Combine like terms: 0 + 15 = 15 7k + k2 = 15 The k term is 7k. Take half its coefficient (3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. 7k + 12.25 + k2 = 15 + 12.25 Reorder the terms: 12.25 + 7k + k2 = 15 + 12.25 Combine like terms: 15 + 12.25 = 27.25 12.25 + 7k + k2 = 27.25 Factor a perfect square on the left side: (k + 3.5)(k + 3.5) = 27.25 Calculate the square root of the right side: 5.220153254 Break this problem into two subproblems by setting (k + 3.5) equal to 5.220153254 and -5.220153254.Subproblem 1
k + 3.5 = 5.220153254 Simplifying k + 3.5 = 5.220153254 Reorder the terms: 3.5 + k = 5.220153254 Solving 3.5 + k = 5.220153254 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + k = 5.220153254 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + k = 5.220153254 + -3.5 k = 5.220153254 + -3.5 Combine like terms: 5.220153254 + -3.5 = 1.720153254 k = 1.720153254 Simplifying k = 1.720153254Subproblem 2
k + 3.5 = -5.220153254 Simplifying k + 3.5 = -5.220153254 Reorder the terms: 3.5 + k = -5.220153254 Solving 3.5 + k = -5.220153254 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + k = -5.220153254 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + k = -5.220153254 + -3.5 k = -5.220153254 + -3.5 Combine like terms: -5.220153254 + -3.5 = -8.720153254 k = -8.720153254 Simplifying k = -8.720153254Solution
The solution to the problem is based on the solutions from the subproblems. k = {1.720153254, -8.720153254}
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