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Simplifying 5k + (-4k2) + -13 + -9k + -1(-7) + 18k2 = 0 Multiply -1 * -7 5k + (-4k2) + -13 + -9k + 7 + 18k2 = 0 Reorder the terms: -13 + 7 + 5k + -9k + (-4k2) + 18k2 = 0 Combine like terms: -13 + 7 = -6 -6 + 5k + -9k + (-4k2) + 18k2 = 0 Combine like terms: 5k + -9k = -4k -6 + -4k + (-4k2) + 18k2 = 0 Combine like terms: (-4k2) + 18k2 = 14k2 -6 + -4k + 14k2 = 0 Solving -6 + -4k + 14k2 = 0 Solving for variable 'k'. Factor out the Greatest Common Factor (GCF), '2'. 2(-3 + -2k + 7k2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-3 + -2k + 7k2)' equal to zero and attempt to solve: Simplifying -3 + -2k + 7k2 = 0 Solving -3 + -2k + 7k2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -0.4285714286 + -0.2857142857k + k2 = 0 Move the constant term to the right: Add '0.4285714286' to each side of the equation. -0.4285714286 + -0.2857142857k + 0.4285714286 + k2 = 0 + 0.4285714286 Reorder the terms: -0.4285714286 + 0.4285714286 + -0.2857142857k + k2 = 0 + 0.4285714286 Combine like terms: -0.4285714286 + 0.4285714286 = 0.0000000000 0.0000000000 + -0.2857142857k + k2 = 0 + 0.4285714286 -0.2857142857k + k2 = 0 + 0.4285714286 Combine like terms: 0 + 0.4285714286 = 0.4285714286 -0.2857142857k + k2 = 0.4285714286 The k term is -0.2857142857k. Take half its coefficient (-0.1428571429). Square it (0.02040816328) and add it to both sides. Add '0.02040816328' to each side of the equation. -0.2857142857k + 0.02040816328 + k2 = 0.4285714286 + 0.02040816328 Reorder the terms: 0.02040816328 + -0.2857142857k + k2 = 0.4285714286 + 0.02040816328 Combine like terms: 0.4285714286 + 0.02040816328 = 0.44897959188 0.02040816328 + -0.2857142857k + k2 = 0.44897959188 Factor a perfect square on the left side: (k + -0.1428571429)(k + -0.1428571429) = 0.44897959188 Calculate the square root of the right side: 0.670059394 Break this problem into two subproblems by setting (k + -0.1428571429) equal to 0.670059394 and -0.670059394.Subproblem 1
k + -0.1428571429 = 0.670059394 Simplifying k + -0.1428571429 = 0.670059394 Reorder the terms: -0.1428571429 + k = 0.670059394 Solving -0.1428571429 + k = 0.670059394 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.1428571429' to each side of the equation. -0.1428571429 + 0.1428571429 + k = 0.670059394 + 0.1428571429 Combine like terms: -0.1428571429 + 0.1428571429 = 0.0000000000 0.0000000000 + k = 0.670059394 + 0.1428571429 k = 0.670059394 + 0.1428571429 Combine like terms: 0.670059394 + 0.1428571429 = 0.8129165369 k = 0.8129165369 Simplifying k = 0.8129165369Subproblem 2
k + -0.1428571429 = -0.670059394 Simplifying k + -0.1428571429 = -0.670059394 Reorder the terms: -0.1428571429 + k = -0.670059394 Solving -0.1428571429 + k = -0.670059394 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.1428571429' to each side of the equation. -0.1428571429 + 0.1428571429 + k = -0.670059394 + 0.1428571429 Combine like terms: -0.1428571429 + 0.1428571429 = 0.0000000000 0.0000000000 + k = -0.670059394 + 0.1428571429 k = -0.670059394 + 0.1428571429 Combine like terms: -0.670059394 + 0.1428571429 = -0.5272022511 k = -0.5272022511 Simplifying k = -0.5272022511Solution
The solution to the problem is based on the solutions from the subproblems. k = {0.8129165369, -0.5272022511}Solution
k = {0.8129165369, -0.5272022511}
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