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Simplifying 4k2 + 14k + 9 = 0 Reorder the terms: 9 + 14k + 4k2 = 0 Solving 9 + 14k + 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 + 3.5k + k2 = 0 Move the constant term to the right: Add '-2.25' to each side of the equation. 2.25 + 3.5k + -2.25 + k2 = 0 + -2.25 Reorder the terms: 2.25 + -2.25 + 3.5k + k2 = 0 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + 3.5k + k2 = 0 + -2.25 3.5k + k2 = 0 + -2.25 Combine like terms: 0 + -2.25 = -2.25 3.5k + k2 = -2.25 The k term is 3.5k. Take half its coefficient (1.75). Square it (3.0625) and add it to both sides. Add '3.0625' to each side of the equation. 3.5k + 3.0625 + k2 = -2.25 + 3.0625 Reorder the terms: 3.0625 + 3.5k + k2 = -2.25 + 3.0625 Combine like terms: -2.25 + 3.0625 = 0.8125 3.0625 + 3.5k + k2 = 0.8125 Factor a perfect square on the left side: (k + 1.75)(k + 1.75) = 0.8125 Calculate the square root of the right side: 0.901387819 Break this problem into two subproblems by setting (k + 1.75) equal to 0.901387819 and -0.901387819.Subproblem 1
k + 1.75 = 0.901387819 Simplifying k + 1.75 = 0.901387819 Reorder the terms: 1.75 + k = 0.901387819 Solving 1.75 + k = 0.901387819 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + k = 0.901387819 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + k = 0.901387819 + -1.75 k = 0.901387819 + -1.75 Combine like terms: 0.901387819 + -1.75 = -0.848612181 k = -0.848612181 Simplifying k = -0.848612181Subproblem 2
k + 1.75 = -0.901387819 Simplifying k + 1.75 = -0.901387819 Reorder the terms: 1.75 + k = -0.901387819 Solving 1.75 + k = -0.901387819 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + k = -0.901387819 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + k = -0.901387819 + -1.75 k = -0.901387819 + -1.75 Combine like terms: -0.901387819 + -1.75 = -2.651387819 k = -2.651387819 Simplifying k = -2.651387819Solution
The solution to the problem is based on the solutions from the subproblems. k = {-0.848612181, -2.651387819}
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