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Simplifying 2k(4 + -3k) + -1(-3) + 2k = 0 (4 * 2k + -3k * 2k) + -1(-3) + 2k = 0 (8k + -6k2) + -1(-3) + 2k = 0 Multiply -1 * -3 8k + -6k2 + 3 + 2k = 0 Reorder the terms: 3 + 8k + 2k + -6k2 = 0 Combine like terms: 8k + 2k = 10k 3 + 10k + -6k2 = 0 Solving 3 + 10k + -6k2 = 0 Solving for variable 'k'. Begin completing the square. Divide all terms by -6 the coefficient of the squared term: Divide each side by '-6'. -0.5 + -1.666666667k + k2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + -1.666666667k + 0.5 + k2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + -1.666666667k + k2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + -1.666666667k + k2 = 0 + 0.5 -1.666666667k + k2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 -1.666666667k + k2 = 0.5 The k term is -1.666666667k. Take half its coefficient (-0.8333333335). Square it (0.6944444447) and add it to both sides. Add '0.6944444447' to each side of the equation. -1.666666667k + 0.6944444447 + k2 = 0.5 + 0.6944444447 Reorder the terms: 0.6944444447 + -1.666666667k + k2 = 0.5 + 0.6944444447 Combine like terms: 0.5 + 0.6944444447 = 1.1944444447 0.6944444447 + -1.666666667k + k2 = 1.1944444447 Factor a perfect square on the left side: (k + -0.8333333335)(k + -0.8333333335) = 1.1944444447 Calculate the square root of the right side: 1.092906421 Break this problem into two subproblems by setting (k + -0.8333333335) equal to 1.092906421 and -1.092906421.Subproblem 1
k + -0.8333333335 = 1.092906421 Simplifying k + -0.8333333335 = 1.092906421 Reorder the terms: -0.8333333335 + k = 1.092906421 Solving -0.8333333335 + k = 1.092906421 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + k = 1.092906421 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + k = 1.092906421 + 0.8333333335 k = 1.092906421 + 0.8333333335 Combine like terms: 1.092906421 + 0.8333333335 = 1.9262397545 k = 1.9262397545 Simplifying k = 1.9262397545Subproblem 2
k + -0.8333333335 = -1.092906421 Simplifying k + -0.8333333335 = -1.092906421 Reorder the terms: -0.8333333335 + k = -1.092906421 Solving -0.8333333335 + k = -1.092906421 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + k = -1.092906421 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + k = -1.092906421 + 0.8333333335 k = -1.092906421 + 0.8333333335 Combine like terms: -1.092906421 + 0.8333333335 = -0.2595730875 k = -0.2595730875 Simplifying k = -0.2595730875Solution
The solution to the problem is based on the solutions from the subproblems. k = {1.9262397545, -0.2595730875}
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