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Simplifying -3 = k2 + 18k + 12 Reorder the terms: -3 = 12 + 18k + k2 Solving -3 = 12 + 18k + k2 Solving for variable 'k'. Combine like terms: -3 + -12 = -15 -15 + -18k + -1k2 = 12 + 18k + k2 + -12 + -18k + -1k2 Reorder the terms: -15 + -18k + -1k2 = 12 + -12 + 18k + -18k + k2 + -1k2 Combine like terms: 12 + -12 = 0 -15 + -18k + -1k2 = 0 + 18k + -18k + k2 + -1k2 -15 + -18k + -1k2 = 18k + -18k + k2 + -1k2 Combine like terms: 18k + -18k = 0 -15 + -18k + -1k2 = 0 + k2 + -1k2 -15 + -18k + -1k2 = k2 + -1k2 Combine like terms: k2 + -1k2 = 0 -15 + -18k + -1k2 = 0 Factor out the Greatest Common Factor (GCF), '-1'. -1(15 + 18k + k2) = 0 Ignore the factor -1.Subproblem 1
Set the factor '(15 + 18k + k2)' equal to zero and attempt to solve: Simplifying 15 + 18k + k2 = 0 Solving 15 + 18k + k2 = 0 Begin completing the square. Move the constant term to the right: Add '-15' to each side of the equation. 15 + 18k + -15 + k2 = 0 + -15 Reorder the terms: 15 + -15 + 18k + k2 = 0 + -15 Combine like terms: 15 + -15 = 0 0 + 18k + k2 = 0 + -15 18k + k2 = 0 + -15 Combine like terms: 0 + -15 = -15 18k + k2 = -15 The k term is 18k. Take half its coefficient (9). Square it (81) and add it to both sides. Add '81' to each side of the equation. 18k + 81 + k2 = -15 + 81 Reorder the terms: 81 + 18k + k2 = -15 + 81 Combine like terms: -15 + 81 = 66 81 + 18k + k2 = 66 Factor a perfect square on the left side: (k + 9)(k + 9) = 66 Calculate the square root of the right side: 8.124038405 Break this problem into two subproblems by setting (k + 9) equal to 8.124038405 and -8.124038405.Subproblem 1
k + 9 = 8.124038405 Simplifying k + 9 = 8.124038405 Reorder the terms: 9 + k = 8.124038405 Solving 9 + k = 8.124038405 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-9' to each side of the equation. 9 + -9 + k = 8.124038405 + -9 Combine like terms: 9 + -9 = 0 0 + k = 8.124038405 + -9 k = 8.124038405 + -9 Combine like terms: 8.124038405 + -9 = -0.875961595 k = -0.875961595 Simplifying k = -0.875961595Subproblem 2
k + 9 = -8.124038405 Simplifying k + 9 = -8.124038405 Reorder the terms: 9 + k = -8.124038405 Solving 9 + k = -8.124038405 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-9' to each side of the equation. 9 + -9 + k = -8.124038405 + -9 Combine like terms: 9 + -9 = 0 0 + k = -8.124038405 + -9 k = -8.124038405 + -9 Combine like terms: -8.124038405 + -9 = -17.124038405 k = -17.124038405 Simplifying k = -17.124038405Solution
The solution to the problem is based on the solutions from the subproblems. k = {-0.875961595, -17.124038405}Solution
k = {-0.875961595, -17.124038405}
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