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Simplifying 9 + -12k + -4k2 = 0 Solving 9 + -12k + -4k2 = 0 Solving for variable 'k'. Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -2.25 + 3k + k2 = 0 Move the constant term to the right: Add '2.25' to each side of the equation. -2.25 + 3k + 2.25 + k2 = 0 + 2.25 Reorder the terms: -2.25 + 2.25 + 3k + k2 = 0 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + 3k + k2 = 0 + 2.25 3k + k2 = 0 + 2.25 Combine like terms: 0 + 2.25 = 2.25 3k + k2 = 2.25 The k term is 3k. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3k + 2.25 + k2 = 2.25 + 2.25 Reorder the terms: 2.25 + 3k + k2 = 2.25 + 2.25 Combine like terms: 2.25 + 2.25 = 4.5 2.25 + 3k + k2 = 4.5 Factor a perfect square on the left side: (k + 1.5)(k + 1.5) = 4.5 Calculate the square root of the right side: 2.121320344 Break this problem into two subproblems by setting (k + 1.5) equal to 2.121320344 and -2.121320344.Subproblem 1
k + 1.5 = 2.121320344 Simplifying k + 1.5 = 2.121320344 Reorder the terms: 1.5 + k = 2.121320344 Solving 1.5 + k = 2.121320344 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + k = 2.121320344 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + k = 2.121320344 + -1.5 k = 2.121320344 + -1.5 Combine like terms: 2.121320344 + -1.5 = 0.621320344 k = 0.621320344 Simplifying k = 0.621320344Subproblem 2
k + 1.5 = -2.121320344 Simplifying k + 1.5 = -2.121320344 Reorder the terms: 1.5 + k = -2.121320344 Solving 1.5 + k = -2.121320344 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + k = -2.121320344 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + k = -2.121320344 + -1.5 k = -2.121320344 + -1.5 Combine like terms: -2.121320344 + -1.5 = -3.621320344 k = -3.621320344 Simplifying k = -3.621320344Solution
The solution to the problem is based on the solutions from the subproblems. k = {0.621320344, -3.621320344}
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