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Simplifying 18k2 + -25k + 3 = 0 Reorder the terms: 3 + -25k + 18k2 = 0 Solving 3 + -25k + 18k2 = 0 Solving for variable 'k'. Begin completing the square. Divide all terms by 18 the coefficient of the squared term: Divide each side by '18'. 0.1666666667 + -1.388888889k + k2 = 0 Move the constant term to the right: Add '-0.1666666667' to each side of the equation. 0.1666666667 + -1.388888889k + -0.1666666667 + k2 = 0 + -0.1666666667 Reorder the terms: 0.1666666667 + -0.1666666667 + -1.388888889k + k2 = 0 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + -1.388888889k + k2 = 0 + -0.1666666667 -1.388888889k + k2 = 0 + -0.1666666667 Combine like terms: 0 + -0.1666666667 = -0.1666666667 -1.388888889k + k2 = -0.1666666667 The k term is -1.388888889k. Take half its coefficient (-0.6944444445). Square it (0.4822530865) and add it to both sides. Add '0.4822530865' to each side of the equation. -1.388888889k + 0.4822530865 + k2 = -0.1666666667 + 0.4822530865 Reorder the terms: 0.4822530865 + -1.388888889k + k2 = -0.1666666667 + 0.4822530865 Combine like terms: -0.1666666667 + 0.4822530865 = 0.3155864198 0.4822530865 + -1.388888889k + k2 = 0.3155864198 Factor a perfect square on the left side: (k + -0.6944444445)(k + -0.6944444445) = 0.3155864198 Calculate the square root of the right side: 0.561770789 Break this problem into two subproblems by setting (k + -0.6944444445) equal to 0.561770789 and -0.561770789.Subproblem 1
k + -0.6944444445 = 0.561770789 Simplifying k + -0.6944444445 = 0.561770789 Reorder the terms: -0.6944444445 + k = 0.561770789 Solving -0.6944444445 + k = 0.561770789 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.6944444445' to each side of the equation. -0.6944444445 + 0.6944444445 + k = 0.561770789 + 0.6944444445 Combine like terms: -0.6944444445 + 0.6944444445 = 0.0000000000 0.0000000000 + k = 0.561770789 + 0.6944444445 k = 0.561770789 + 0.6944444445 Combine like terms: 0.561770789 + 0.6944444445 = 1.2562152335 k = 1.2562152335 Simplifying k = 1.2562152335Subproblem 2
k + -0.6944444445 = -0.561770789 Simplifying k + -0.6944444445 = -0.561770789 Reorder the terms: -0.6944444445 + k = -0.561770789 Solving -0.6944444445 + k = -0.561770789 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.6944444445' to each side of the equation. -0.6944444445 + 0.6944444445 + k = -0.561770789 + 0.6944444445 Combine like terms: -0.6944444445 + 0.6944444445 = 0.0000000000 0.0000000000 + k = -0.561770789 + 0.6944444445 k = -0.561770789 + 0.6944444445 Combine like terms: -0.561770789 + 0.6944444445 = 0.1326736555 k = 0.1326736555 Simplifying k = 0.1326736555Solution
The solution to the problem is based on the solutions from the subproblems. k = {1.2562152335, 0.1326736555}
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