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Simplifying 17k2 + -28k + -24 = 3k2 Reorder the terms: -24 + -28k + 17k2 = 3k2 Solving -24 + -28k + 17k2 = 3k2 Solving for variable 'k'. Combine like terms: 17k2 + -3k2 = 14k2 -24 + -28k + 14k2 = 3k2 + -3k2 Combine like terms: 3k2 + -3k2 = 0 -24 + -28k + 14k2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-12 + -14k + 7k2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-12 + -14k + 7k2)' equal to zero and attempt to solve: Simplifying -12 + -14k + 7k2 = 0 Solving -12 + -14k + 7k2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -1.714285714 + -2k + k2 = 0 Move the constant term to the right: Add '1.714285714' to each side of the equation. -1.714285714 + -2k + 1.714285714 + k2 = 0 + 1.714285714 Reorder the terms: -1.714285714 + 1.714285714 + -2k + k2 = 0 + 1.714285714 Combine like terms: -1.714285714 + 1.714285714 = 0.000000000 0.000000000 + -2k + k2 = 0 + 1.714285714 -2k + k2 = 0 + 1.714285714 Combine like terms: 0 + 1.714285714 = 1.714285714 -2k + k2 = 1.714285714 The k term is -2k. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2k + 1 + k2 = 1.714285714 + 1 Reorder the terms: 1 + -2k + k2 = 1.714285714 + 1 Combine like terms: 1.714285714 + 1 = 2.714285714 1 + -2k + k2 = 2.714285714 Factor a perfect square on the left side: (k + -1)(k + -1) = 2.714285714 Calculate the square root of the right side: 1.647508942 Break this problem into two subproblems by setting (k + -1) equal to 1.647508942 and -1.647508942.Subproblem 1
k + -1 = 1.647508942 Simplifying k + -1 = 1.647508942 Reorder the terms: -1 + k = 1.647508942 Solving -1 + k = 1.647508942 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + k = 1.647508942 + 1 Combine like terms: -1 + 1 = 0 0 + k = 1.647508942 + 1 k = 1.647508942 + 1 Combine like terms: 1.647508942 + 1 = 2.647508942 k = 2.647508942 Simplifying k = 2.647508942Subproblem 2
k + -1 = -1.647508942 Simplifying k + -1 = -1.647508942 Reorder the terms: -1 + k = -1.647508942 Solving -1 + k = -1.647508942 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + k = -1.647508942 + 1 Combine like terms: -1 + 1 = 0 0 + k = -1.647508942 + 1 k = -1.647508942 + 1 Combine like terms: -1.647508942 + 1 = -0.647508942 k = -0.647508942 Simplifying k = -0.647508942Solution
The solution to the problem is based on the solutions from the subproblems. k = {2.647508942, -0.647508942}Solution
k = {2.647508942, -0.647508942}
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