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Simplifying 2k3 + 6k2 + -14k = 0 Reorder the terms: -14k + 6k2 + 2k3 = 0 Solving -14k + 6k2 + 2k3 = 0 Solving for variable 'k'. Factor out the Greatest Common Factor (GCF), '2k'. 2k(-7 + 3k + k2) = 0 Ignore the factor 2.Subproblem 1
Set the factor 'k' equal to zero and attempt to solve: Simplifying k = 0 Solving k = 0 Move all terms containing k to the left, all other terms to the right. Simplifying k = 0Subproblem 2
Set the factor '(-7 + 3k + k2)' equal to zero and attempt to solve: Simplifying -7 + 3k + k2 = 0 Solving -7 + 3k + k2 = 0 Begin completing the square. Move the constant term to the right: Add '7' to each side of the equation. -7 + 3k + 7 + k2 = 0 + 7 Reorder the terms: -7 + 7 + 3k + k2 = 0 + 7 Combine like terms: -7 + 7 = 0 0 + 3k + k2 = 0 + 7 3k + k2 = 0 + 7 Combine like terms: 0 + 7 = 7 3k + k2 = 7 The k term is 3k. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3k + 2.25 + k2 = 7 + 2.25 Reorder the terms: 2.25 + 3k + k2 = 7 + 2.25 Combine like terms: 7 + 2.25 = 9.25 2.25 + 3k + k2 = 9.25 Factor a perfect square on the left side: (k + 1.5)(k + 1.5) = 9.25 Calculate the square root of the right side: 3.041381265 Break this problem into two subproblems by setting (k + 1.5) equal to 3.041381265 and -3.041381265.Subproblem 1
k + 1.5 = 3.041381265 Simplifying k + 1.5 = 3.041381265 Reorder the terms: 1.5 + k = 3.041381265 Solving 1.5 + k = 3.041381265 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + k = 3.041381265 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + k = 3.041381265 + -1.5 k = 3.041381265 + -1.5 Combine like terms: 3.041381265 + -1.5 = 1.541381265 k = 1.541381265 Simplifying k = 1.541381265Subproblem 2
k + 1.5 = -3.041381265 Simplifying k + 1.5 = -3.041381265 Reorder the terms: 1.5 + k = -3.041381265 Solving 1.5 + k = -3.041381265 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + k = -3.041381265 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + k = -3.041381265 + -1.5 k = -3.041381265 + -1.5 Combine like terms: -3.041381265 + -1.5 = -4.541381265 k = -4.541381265 Simplifying k = -4.541381265Solution
The solution to the problem is based on the solutions from the subproblems. k = {1.541381265, -4.541381265}Solution
k = {0, 1.541381265, -4.541381265}
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