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Simplifying 7k2 + -6k * 3 = 3 Reorder the terms for easier multiplication: 7k2 + -6 * 3k = 3 Multiply -6 * 3 7k2 + -18k = 3 Reorder the terms: -18k + 7k2 = 3 Solving -18k + 7k2 = 3 Solving for variable 'k'. Reorder the terms: -3 + -18k + 7k2 = 3 + -3 Combine like terms: 3 + -3 = 0 -3 + -18k + 7k2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -0.4285714286 + -2.571428571k + k2 = 0 Move the constant term to the right: Add '0.4285714286' to each side of the equation. -0.4285714286 + -2.571428571k + 0.4285714286 + k2 = 0 + 0.4285714286 Reorder the terms: -0.4285714286 + 0.4285714286 + -2.571428571k + k2 = 0 + 0.4285714286 Combine like terms: -0.4285714286 + 0.4285714286 = 0.0000000000 0.0000000000 + -2.571428571k + k2 = 0 + 0.4285714286 -2.571428571k + k2 = 0 + 0.4285714286 Combine like terms: 0 + 0.4285714286 = 0.4285714286 -2.571428571k + k2 = 0.4285714286 The k term is -2.571428571k. Take half its coefficient (-1.285714286). Square it (1.653061225) and add it to both sides. Add '1.653061225' to each side of the equation. -2.571428571k + 1.653061225 + k2 = 0.4285714286 + 1.653061225 Reorder the terms: 1.653061225 + -2.571428571k + k2 = 0.4285714286 + 1.653061225 Combine like terms: 0.4285714286 + 1.653061225 = 2.0816326536 1.653061225 + -2.571428571k + k2 = 2.0816326536 Factor a perfect square on the left side: (k + -1.285714286)(k + -1.285714286) = 2.0816326536 Calculate the square root of the right side: 1.44278642 Break this problem into two subproblems by setting (k + -1.285714286) equal to 1.44278642 and -1.44278642.Subproblem 1
k + -1.285714286 = 1.44278642 Simplifying k + -1.285714286 = 1.44278642 Reorder the terms: -1.285714286 + k = 1.44278642 Solving -1.285714286 + k = 1.44278642 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '1.285714286' to each side of the equation. -1.285714286 + 1.285714286 + k = 1.44278642 + 1.285714286 Combine like terms: -1.285714286 + 1.285714286 = 0.000000000 0.000000000 + k = 1.44278642 + 1.285714286 k = 1.44278642 + 1.285714286 Combine like terms: 1.44278642 + 1.285714286 = 2.728500706 k = 2.728500706 Simplifying k = 2.728500706Subproblem 2
k + -1.285714286 = -1.44278642 Simplifying k + -1.285714286 = -1.44278642 Reorder the terms: -1.285714286 + k = -1.44278642 Solving -1.285714286 + k = -1.44278642 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '1.285714286' to each side of the equation. -1.285714286 + 1.285714286 + k = -1.44278642 + 1.285714286 Combine like terms: -1.285714286 + 1.285714286 = 0.000000000 0.000000000 + k = -1.44278642 + 1.285714286 k = -1.44278642 + 1.285714286 Combine like terms: -1.44278642 + 1.285714286 = -0.157072134 k = -0.157072134 Simplifying k = -0.157072134Solution
The solution to the problem is based on the solutions from the subproblems. k = {2.728500706, -0.157072134}
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