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Simplifying -36k2 + 16k + 28 = 0 Reorder the terms: 28 + 16k + -36k2 = 0 Solving 28 + 16k + -36k2 = 0 Solving for variable 'k'. Factor out the Greatest Common Factor (GCF), '4'. 4(7 + 4k + -9k2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(7 + 4k + -9k2)' equal to zero and attempt to solve: Simplifying 7 + 4k + -9k2 = 0 Solving 7 + 4k + -9k2 = 0 Begin completing the square. Divide all terms by -9 the coefficient of the squared term: Divide each side by '-9'. -0.7777777778 + -0.4444444444k + k2 = 0 Move the constant term to the right: Add '0.7777777778' to each side of the equation. -0.7777777778 + -0.4444444444k + 0.7777777778 + k2 = 0 + 0.7777777778 Reorder the terms: -0.7777777778 + 0.7777777778 + -0.4444444444k + k2 = 0 + 0.7777777778 Combine like terms: -0.7777777778 + 0.7777777778 = 0.0000000000 0.0000000000 + -0.4444444444k + k2 = 0 + 0.7777777778 -0.4444444444k + k2 = 0 + 0.7777777778 Combine like terms: 0 + 0.7777777778 = 0.7777777778 -0.4444444444k + k2 = 0.7777777778 The k term is -0.4444444444k. Take half its coefficient (-0.2222222222). Square it (0.04938271604) and add it to both sides. Add '0.04938271604' to each side of the equation. -0.4444444444k + 0.04938271604 + k2 = 0.7777777778 + 0.04938271604 Reorder the terms: 0.04938271604 + -0.4444444444k + k2 = 0.7777777778 + 0.04938271604 Combine like terms: 0.7777777778 + 0.04938271604 = 0.82716049384 0.04938271604 + -0.4444444444k + k2 = 0.82716049384 Factor a perfect square on the left side: (k + -0.2222222222)(k + -0.2222222222) = 0.82716049384 Calculate the square root of the right side: 0.909483641 Break this problem into two subproblems by setting (k + -0.2222222222) equal to 0.909483641 and -0.909483641.Subproblem 1
k + -0.2222222222 = 0.909483641 Simplifying k + -0.2222222222 = 0.909483641 Reorder the terms: -0.2222222222 + k = 0.909483641 Solving -0.2222222222 + k = 0.909483641 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.2222222222' to each side of the equation. -0.2222222222 + 0.2222222222 + k = 0.909483641 + 0.2222222222 Combine like terms: -0.2222222222 + 0.2222222222 = 0.0000000000 0.0000000000 + k = 0.909483641 + 0.2222222222 k = 0.909483641 + 0.2222222222 Combine like terms: 0.909483641 + 0.2222222222 = 1.1317058632 k = 1.1317058632 Simplifying k = 1.1317058632Subproblem 2
k + -0.2222222222 = -0.909483641 Simplifying k + -0.2222222222 = -0.909483641 Reorder the terms: -0.2222222222 + k = -0.909483641 Solving -0.2222222222 + k = -0.909483641 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.2222222222' to each side of the equation. -0.2222222222 + 0.2222222222 + k = -0.909483641 + 0.2222222222 Combine like terms: -0.2222222222 + 0.2222222222 = 0.0000000000 0.0000000000 + k = -0.909483641 + 0.2222222222 k = -0.909483641 + 0.2222222222 Combine like terms: -0.909483641 + 0.2222222222 = -0.6872614188 k = -0.6872614188 Simplifying k = -0.6872614188Solution
The solution to the problem is based on the solutions from the subproblems. k = {1.1317058632, -0.6872614188}Solution
k = {1.1317058632, -0.6872614188}
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