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Simplifying 3k2 + 48k = -123 Reorder the terms: 48k + 3k2 = -123 Solving 48k + 3k2 = -123 Solving for variable 'k'. Reorder the terms: 123 + 48k + 3k2 = -123 + 123 Combine like terms: -123 + 123 = 0 123 + 48k + 3k2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(41 + 16k + k2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(41 + 16k + k2)' equal to zero and attempt to solve: Simplifying 41 + 16k + k2 = 0 Solving 41 + 16k + k2 = 0 Begin completing the square. Move the constant term to the right: Add '-41' to each side of the equation. 41 + 16k + -41 + k2 = 0 + -41 Reorder the terms: 41 + -41 + 16k + k2 = 0 + -41 Combine like terms: 41 + -41 = 0 0 + 16k + k2 = 0 + -41 16k + k2 = 0 + -41 Combine like terms: 0 + -41 = -41 16k + k2 = -41 The k term is 16k. Take half its coefficient (8). Square it (64) and add it to both sides. Add '64' to each side of the equation. 16k + 64 + k2 = -41 + 64 Reorder the terms: 64 + 16k + k2 = -41 + 64 Combine like terms: -41 + 64 = 23 64 + 16k + k2 = 23 Factor a perfect square on the left side: (k + 8)(k + 8) = 23 Calculate the square root of the right side: 4.795831523 Break this problem into two subproblems by setting (k + 8) equal to 4.795831523 and -4.795831523.Subproblem 1
k + 8 = 4.795831523 Simplifying k + 8 = 4.795831523 Reorder the terms: 8 + k = 4.795831523 Solving 8 + k = 4.795831523 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-8' to each side of the equation. 8 + -8 + k = 4.795831523 + -8 Combine like terms: 8 + -8 = 0 0 + k = 4.795831523 + -8 k = 4.795831523 + -8 Combine like terms: 4.795831523 + -8 = -3.204168477 k = -3.204168477 Simplifying k = -3.204168477Subproblem 2
k + 8 = -4.795831523 Simplifying k + 8 = -4.795831523 Reorder the terms: 8 + k = -4.795831523 Solving 8 + k = -4.795831523 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-8' to each side of the equation. 8 + -8 + k = -4.795831523 + -8 Combine like terms: 8 + -8 = 0 0 + k = -4.795831523 + -8 k = -4.795831523 + -8 Combine like terms: -4.795831523 + -8 = -12.795831523 k = -12.795831523 Simplifying k = -12.795831523Solution
The solution to the problem is based on the solutions from the subproblems. k = {-3.204168477, -12.795831523}Solution
k = {-3.204168477, -12.795831523}
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