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Simplifying 4k2 + 9k + -4 = 0 Reorder the terms: -4 + 9k + 4k2 = 0 Solving -4 + 9k + 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'. -1 + 2.25k + k2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + 2.25k + 1 + k2 = 0 + 1 Reorder the terms: -1 + 1 + 2.25k + k2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + 2.25k + k2 = 0 + 1 2.25k + k2 = 0 + 1 Combine like terms: 0 + 1 = 1 2.25k + k2 = 1 The k term is 2.25k. Take half its coefficient (1.125). Square it (1.265625) and add it to both sides. Add '1.265625' to each side of the equation. 2.25k + 1.265625 + k2 = 1 + 1.265625 Reorder the terms: 1.265625 + 2.25k + k2 = 1 + 1.265625 Combine like terms: 1 + 1.265625 = 2.265625 1.265625 + 2.25k + k2 = 2.265625 Factor a perfect square on the left side: (k + 1.125)(k + 1.125) = 2.265625 Calculate the square root of the right side: 1.505199322 Break this problem into two subproblems by setting (k + 1.125) equal to 1.505199322 and -1.505199322.Subproblem 1
k + 1.125 = 1.505199322 Simplifying k + 1.125 = 1.505199322 Reorder the terms: 1.125 + k = 1.505199322 Solving 1.125 + k = 1.505199322 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-1.125' to each side of the equation. 1.125 + -1.125 + k = 1.505199322 + -1.125 Combine like terms: 1.125 + -1.125 = 0.000 0.000 + k = 1.505199322 + -1.125 k = 1.505199322 + -1.125 Combine like terms: 1.505199322 + -1.125 = 0.380199322 k = 0.380199322 Simplifying k = 0.380199322Subproblem 2
k + 1.125 = -1.505199322 Simplifying k + 1.125 = -1.505199322 Reorder the terms: 1.125 + k = -1.505199322 Solving 1.125 + k = -1.505199322 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-1.125' to each side of the equation. 1.125 + -1.125 + k = -1.505199322 + -1.125 Combine like terms: 1.125 + -1.125 = 0.000 0.000 + k = -1.505199322 + -1.125 k = -1.505199322 + -1.125 Combine like terms: -1.505199322 + -1.125 = -2.630199322 k = -2.630199322 Simplifying k = -2.630199322Solution
The solution to the problem is based on the solutions from the subproblems. k = {0.380199322, -2.630199322}
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