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Simplifying 2k2 + 9k + -9 = 0 Reorder the terms: -9 + 9k + 2k2 = 0 Solving -9 + 9k + 2k2 = 0 Solving for variable 'k'. Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -4.5 + 4.5k + k2 = 0 Move the constant term to the right: Add '4.5' to each side of the equation. -4.5 + 4.5k + 4.5 + k2 = 0 + 4.5 Reorder the terms: -4.5 + 4.5 + 4.5k + k2 = 0 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + 4.5k + k2 = 0 + 4.5 4.5k + k2 = 0 + 4.5 Combine like terms: 0 + 4.5 = 4.5 4.5k + k2 = 4.5 The k term is 4.5k. Take half its coefficient (2.25). Square it (5.0625) and add it to both sides. Add '5.0625' to each side of the equation. 4.5k + 5.0625 + k2 = 4.5 + 5.0625 Reorder the terms: 5.0625 + 4.5k + k2 = 4.5 + 5.0625 Combine like terms: 4.5 + 5.0625 = 9.5625 5.0625 + 4.5k + k2 = 9.5625 Factor a perfect square on the left side: (k + 2.25)(k + 2.25) = 9.5625 Calculate the square root of the right side: 3.092329219 Break this problem into two subproblems by setting (k + 2.25) equal to 3.092329219 and -3.092329219.Subproblem 1
k + 2.25 = 3.092329219 Simplifying k + 2.25 = 3.092329219 Reorder the terms: 2.25 + k = 3.092329219 Solving 2.25 + k = 3.092329219 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-2.25' to each side of the equation. 2.25 + -2.25 + k = 3.092329219 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + k = 3.092329219 + -2.25 k = 3.092329219 + -2.25 Combine like terms: 3.092329219 + -2.25 = 0.842329219 k = 0.842329219 Simplifying k = 0.842329219Subproblem 2
k + 2.25 = -3.092329219 Simplifying k + 2.25 = -3.092329219 Reorder the terms: 2.25 + k = -3.092329219 Solving 2.25 + k = -3.092329219 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-2.25' to each side of the equation. 2.25 + -2.25 + k = -3.092329219 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + k = -3.092329219 + -2.25 k = -3.092329219 + -2.25 Combine like terms: -3.092329219 + -2.25 = -5.342329219 k = -5.342329219 Simplifying k = -5.342329219Solution
The solution to the problem is based on the solutions from the subproblems. k = {0.842329219, -5.342329219}
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