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Simplifying k + 4k + 4 = k * k Reorder the terms: 4 + k + 4k = k * k Combine like terms: k + 4k = 5k 4 + 5k = k * k Multiply k * k 4 + 5k = k2 Solving 4 + 5k = k2 Solving for variable 'k'. Combine like terms: k2 + -1k2 = 0 4 + 5k + -1k2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -4 + -5k + k2 = 0 Move the constant term to the right: Add '4' to each side of the equation. -4 + -5k + 4 + k2 = 0 + 4 Reorder the terms: -4 + 4 + -5k + k2 = 0 + 4 Combine like terms: -4 + 4 = 0 0 + -5k + k2 = 0 + 4 -5k + k2 = 0 + 4 Combine like terms: 0 + 4 = 4 -5k + k2 = 4 The k term is -5k. Take half its coefficient (-2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. -5k + 6.25 + k2 = 4 + 6.25 Reorder the terms: 6.25 + -5k + k2 = 4 + 6.25 Combine like terms: 4 + 6.25 = 10.25 6.25 + -5k + k2 = 10.25 Factor a perfect square on the left side: (k + -2.5)(k + -2.5) = 10.25 Calculate the square root of the right side: 3.201562119 Break this problem into two subproblems by setting (k + -2.5) equal to 3.201562119 and -3.201562119.Subproblem 1
k + -2.5 = 3.201562119 Simplifying k + -2.5 = 3.201562119 Reorder the terms: -2.5 + k = 3.201562119 Solving -2.5 + k = 3.201562119 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + k = 3.201562119 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + k = 3.201562119 + 2.5 k = 3.201562119 + 2.5 Combine like terms: 3.201562119 + 2.5 = 5.701562119 k = 5.701562119 Simplifying k = 5.701562119Subproblem 2
k + -2.5 = -3.201562119 Simplifying k + -2.5 = -3.201562119 Reorder the terms: -2.5 + k = -3.201562119 Solving -2.5 + k = -3.201562119 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + k = -3.201562119 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + k = -3.201562119 + 2.5 k = -3.201562119 + 2.5 Combine like terms: -3.201562119 + 2.5 = -0.701562119 k = -0.701562119 Simplifying k = -0.701562119Solution
The solution to the problem is based on the solutions from the subproblems. k = {5.701562119, -0.701562119}
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