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Simplifying (2k2 + -1k + 3) + (5k2 + 3k + -7) = 0 Reorder the terms: (3 + -1k + 2k2) + (5k2 + 3k + -7) = 0 Remove parenthesis around (3 + -1k + 2k2) 3 + -1k + 2k2 + (5k2 + 3k + -7) = 0 Reorder the terms: 3 + -1k + 2k2 + (-7 + 3k + 5k2) = 0 Remove parenthesis around (-7 + 3k + 5k2) 3 + -1k + 2k2 + -7 + 3k + 5k2 = 0 Reorder the terms: 3 + -7 + -1k + 3k + 2k2 + 5k2 = 0 Combine like terms: 3 + -7 = -4 -4 + -1k + 3k + 2k2 + 5k2 = 0 Combine like terms: -1k + 3k = 2k -4 + 2k + 2k2 + 5k2 = 0 Combine like terms: 2k2 + 5k2 = 7k2 -4 + 2k + 7k2 = 0 Solving -4 + 2k + 7k2 = 0 Solving for variable 'k'. Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -0.5714285714 + 0.2857142857k + k2 = 0 Move the constant term to the right: Add '0.5714285714' to each side of the equation. -0.5714285714 + 0.2857142857k + 0.5714285714 + k2 = 0 + 0.5714285714 Reorder the terms: -0.5714285714 + 0.5714285714 + 0.2857142857k + k2 = 0 + 0.5714285714 Combine like terms: -0.5714285714 + 0.5714285714 = 0.0000000000 0.0000000000 + 0.2857142857k + k2 = 0 + 0.5714285714 0.2857142857k + k2 = 0 + 0.5714285714 Combine like terms: 0 + 0.5714285714 = 0.5714285714 0.2857142857k + k2 = 0.5714285714 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 = 0.5714285714 + 0.02040816328 Reorder the terms: 0.02040816328 + 0.2857142857k + k2 = 0.5714285714 + 0.02040816328 Combine like terms: 0.5714285714 + 0.02040816328 = 0.59183673468 0.02040816328 + 0.2857142857k + k2 = 0.59183673468 Factor a perfect square on the left side: (k + 0.1428571429)(k + 0.1428571429) = 0.59183673468 Calculate the square root of the right side: 0.769309258 Break this problem into two subproblems by setting (k + 0.1428571429) equal to 0.769309258 and -0.769309258.Subproblem 1
k + 0.1428571429 = 0.769309258 Simplifying k + 0.1428571429 = 0.769309258 Reorder the terms: 0.1428571429 + k = 0.769309258 Solving 0.1428571429 + k = 0.769309258 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 = 0.769309258 + -0.1428571429 Combine like terms: 0.1428571429 + -0.1428571429 = 0.0000000000 0.0000000000 + k = 0.769309258 + -0.1428571429 k = 0.769309258 + -0.1428571429 Combine like terms: 0.769309258 + -0.1428571429 = 0.6264521151 k = 0.6264521151 Simplifying k = 0.6264521151Subproblem 2
k + 0.1428571429 = -0.769309258 Simplifying k + 0.1428571429 = -0.769309258 Reorder the terms: 0.1428571429 + k = -0.769309258 Solving 0.1428571429 + k = -0.769309258 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 = -0.769309258 + -0.1428571429 Combine like terms: 0.1428571429 + -0.1428571429 = 0.0000000000 0.0000000000 + k = -0.769309258 + -0.1428571429 k = -0.769309258 + -0.1428571429 Combine like terms: -0.769309258 + -0.1428571429 = -0.9121664009 k = -0.9121664009 Simplifying k = -0.9121664009Solution
The solution to the problem is based on the solutions from the subproblems. k = {0.6264521151, -0.9121664009}
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