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Simplifying 6k2 + -4k + -9 = 0 Reorder the terms: -9 + -4k + 6k2 = 0 Solving -9 + -4k + 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'. -1.5 + -0.6666666667k + k2 = 0 Move the constant term to the right: Add '1.5' to each side of the equation. -1.5 + -0.6666666667k + 1.5 + k2 = 0 + 1.5 Reorder the terms: -1.5 + 1.5 + -0.6666666667k + k2 = 0 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + -0.6666666667k + k2 = 0 + 1.5 -0.6666666667k + k2 = 0 + 1.5 Combine like terms: 0 + 1.5 = 1.5 -0.6666666667k + k2 = 1.5 The k term is -0.6666666667k. Take half its coefficient (-0.3333333334). Square it (0.1111111112) and add it to both sides. Add '0.1111111112' to each side of the equation. -0.6666666667k + 0.1111111112 + k2 = 1.5 + 0.1111111112 Reorder the terms: 0.1111111112 + -0.6666666667k + k2 = 1.5 + 0.1111111112 Combine like terms: 1.5 + 0.1111111112 = 1.6111111112 0.1111111112 + -0.6666666667k + k2 = 1.6111111112 Factor a perfect square on the left side: (k + -0.3333333334)(k + -0.3333333334) = 1.6111111112 Calculate the square root of the right side: 1.269295518 Break this problem into two subproblems by setting (k + -0.3333333334) equal to 1.269295518 and -1.269295518.Subproblem 1
k + -0.3333333334 = 1.269295518 Simplifying k + -0.3333333334 = 1.269295518 Reorder the terms: -0.3333333334 + k = 1.269295518 Solving -0.3333333334 + k = 1.269295518 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + k = 1.269295518 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + k = 1.269295518 + 0.3333333334 k = 1.269295518 + 0.3333333334 Combine like terms: 1.269295518 + 0.3333333334 = 1.6026288514 k = 1.6026288514 Simplifying k = 1.6026288514Subproblem 2
k + -0.3333333334 = -1.269295518 Simplifying k + -0.3333333334 = -1.269295518 Reorder the terms: -0.3333333334 + k = -1.269295518 Solving -0.3333333334 + k = -1.269295518 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + k = -1.269295518 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + k = -1.269295518 + 0.3333333334 k = -1.269295518 + 0.3333333334 Combine like terms: -1.269295518 + 0.3333333334 = -0.9359621846 k = -0.9359621846 Simplifying k = -0.9359621846Solution
The solution to the problem is based on the solutions from the subproblems. k = {1.6026288514, -0.9359621846}
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