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Simplifying -16k2 + 18k + 24 = 0 Reorder the terms: 24 + 18k + -16k2 = 0 Solving 24 + 18k + -16k2 = 0 Solving for variable 'k'. Factor out the Greatest Common Factor (GCF), '2'. 2(12 + 9k + -8k2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(12 + 9k + -8k2)' equal to zero and attempt to solve: Simplifying 12 + 9k + -8k2 = 0 Solving 12 + 9k + -8k2 = 0 Begin completing the square. Divide all terms by -8 the coefficient of the squared term: Divide each side by '-8'. -1.5 + -1.125k + k2 = 0 Move the constant term to the right: Add '1.5' to each side of the equation. -1.5 + -1.125k + 1.5 + k2 = 0 + 1.5 Reorder the terms: -1.5 + 1.5 + -1.125k + k2 = 0 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + -1.125k + k2 = 0 + 1.5 -1.125k + k2 = 0 + 1.5 Combine like terms: 0 + 1.5 = 1.5 -1.125k + k2 = 1.5 The k term is -1.125k. Take half its coefficient (-0.5625). Square it (0.31640625) and add it to both sides. Add '0.31640625' to each side of the equation. -1.125k + 0.31640625 + k2 = 1.5 + 0.31640625 Reorder the terms: 0.31640625 + -1.125k + k2 = 1.5 + 0.31640625 Combine like terms: 1.5 + 0.31640625 = 1.81640625 0.31640625 + -1.125k + k2 = 1.81640625 Factor a perfect square on the left side: (k + -0.5625)(k + -0.5625) = 1.81640625 Calculate the square root of the right side: 1.347741166 Break this problem into two subproblems by setting (k + -0.5625) equal to 1.347741166 and -1.347741166.Subproblem 1
k + -0.5625 = 1.347741166 Simplifying k + -0.5625 = 1.347741166 Reorder the terms: -0.5625 + k = 1.347741166 Solving -0.5625 + k = 1.347741166 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.5625' to each side of the equation. -0.5625 + 0.5625 + k = 1.347741166 + 0.5625 Combine like terms: -0.5625 + 0.5625 = 0.0000 0.0000 + k = 1.347741166 + 0.5625 k = 1.347741166 + 0.5625 Combine like terms: 1.347741166 + 0.5625 = 1.910241166 k = 1.910241166 Simplifying k = 1.910241166Subproblem 2
k + -0.5625 = -1.347741166 Simplifying k + -0.5625 = -1.347741166 Reorder the terms: -0.5625 + k = -1.347741166 Solving -0.5625 + k = -1.347741166 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.5625' to each side of the equation. -0.5625 + 0.5625 + k = -1.347741166 + 0.5625 Combine like terms: -0.5625 + 0.5625 = 0.0000 0.0000 + k = -1.347741166 + 0.5625 k = -1.347741166 + 0.5625 Combine like terms: -1.347741166 + 0.5625 = -0.785241166 k = -0.785241166 Simplifying k = -0.785241166Solution
The solution to the problem is based on the solutions from the subproblems. k = {1.910241166, -0.785241166}Solution
k = {1.910241166, -0.785241166}
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