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Simplifying 3k2 + -30k = 111 Reorder the terms: -30k + 3k2 = 111 Solving -30k + 3k2 = 111 Solving for variable 'k'. Reorder the terms: -111 + -30k + 3k2 = 111 + -111 Combine like terms: 111 + -111 = 0 -111 + -30k + 3k2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(-37 + -10k + k2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-37 + -10k + k2)' equal to zero and attempt to solve: Simplifying -37 + -10k + k2 = 0 Solving -37 + -10k + k2 = 0 Begin completing the square. Move the constant term to the right: Add '37' to each side of the equation. -37 + -10k + 37 + k2 = 0 + 37 Reorder the terms: -37 + 37 + -10k + k2 = 0 + 37 Combine like terms: -37 + 37 = 0 0 + -10k + k2 = 0 + 37 -10k + k2 = 0 + 37 Combine like terms: 0 + 37 = 37 -10k + k2 = 37 The k term is -10k. Take half its coefficient (-5). Square it (25) and add it to both sides. Add '25' to each side of the equation. -10k + 25 + k2 = 37 + 25 Reorder the terms: 25 + -10k + k2 = 37 + 25 Combine like terms: 37 + 25 = 62 25 + -10k + k2 = 62 Factor a perfect square on the left side: (k + -5)(k + -5) = 62 Calculate the square root of the right side: 7.874007874 Break this problem into two subproblems by setting (k + -5) equal to 7.874007874 and -7.874007874.Subproblem 1
k + -5 = 7.874007874 Simplifying k + -5 = 7.874007874 Reorder the terms: -5 + k = 7.874007874 Solving -5 + k = 7.874007874 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + k = 7.874007874 + 5 Combine like terms: -5 + 5 = 0 0 + k = 7.874007874 + 5 k = 7.874007874 + 5 Combine like terms: 7.874007874 + 5 = 12.874007874 k = 12.874007874 Simplifying k = 12.874007874Subproblem 2
k + -5 = -7.874007874 Simplifying k + -5 = -7.874007874 Reorder the terms: -5 + k = -7.874007874 Solving -5 + k = -7.874007874 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + k = -7.874007874 + 5 Combine like terms: -5 + 5 = 0 0 + k = -7.874007874 + 5 k = -7.874007874 + 5 Combine like terms: -7.874007874 + 5 = -2.874007874 k = -2.874007874 Simplifying k = -2.874007874Solution
The solution to the problem is based on the solutions from the subproblems. k = {12.874007874, -2.874007874}Solution
k = {12.874007874, -2.874007874}
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