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Simplifying -4k4 + -20k3 + 23k2 = 0 Reorder the terms: 23k2 + -20k3 + -4k4 = 0 Solving 23k2 + -20k3 + -4k4 = 0 Solving for variable 'k'. Factor out the Greatest Common Factor (GCF), 'k2'. k2(23 + -20k + -4k2) = 0Subproblem 1
Set the factor 'k2' equal to zero and attempt to solve: Simplifying k2 = 0 Solving k2 = 0 Move all terms containing k to the left, all other terms to the right. Simplifying k2 = 0 Take the square root of each side: k = {0}Subproblem 2
Set the factor '(23 + -20k + -4k2)' equal to zero and attempt to solve: Simplifying 23 + -20k + -4k2 = 0 Solving 23 + -20k + -4k2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -5.75 + 5k + k2 = 0 Move the constant term to the right: Add '5.75' to each side of the equation. -5.75 + 5k + 5.75 + k2 = 0 + 5.75 Reorder the terms: -5.75 + 5.75 + 5k + k2 = 0 + 5.75 Combine like terms: -5.75 + 5.75 = 0.00 0.00 + 5k + k2 = 0 + 5.75 5k + k2 = 0 + 5.75 Combine like terms: 0 + 5.75 = 5.75 5k + k2 = 5.75 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 = 5.75 + 6.25 Reorder the terms: 6.25 + 5k + k2 = 5.75 + 6.25 Combine like terms: 5.75 + 6.25 = 12 6.25 + 5k + k2 = 12 Factor a perfect square on the left side: (k + 2.5)(k + 2.5) = 12 Calculate the square root of the right side: 3.464101615 Break this problem into two subproblems by setting (k + 2.5) equal to 3.464101615 and -3.464101615.Subproblem 1
k + 2.5 = 3.464101615 Simplifying k + 2.5 = 3.464101615 Reorder the terms: 2.5 + k = 3.464101615 Solving 2.5 + k = 3.464101615 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 = 3.464101615 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + k = 3.464101615 + -2.5 k = 3.464101615 + -2.5 Combine like terms: 3.464101615 + -2.5 = 0.964101615 k = 0.964101615 Simplifying k = 0.964101615Subproblem 2
k + 2.5 = -3.464101615 Simplifying k + 2.5 = -3.464101615 Reorder the terms: 2.5 + k = -3.464101615 Solving 2.5 + k = -3.464101615 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 = -3.464101615 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + k = -3.464101615 + -2.5 k = -3.464101615 + -2.5 Combine like terms: -3.464101615 + -2.5 = -5.964101615 k = -5.964101615 Simplifying k = -5.964101615Solution
The solution to the problem is based on the solutions from the subproblems. k = {0.964101615, -5.964101615}Solution
k = {0, 0.964101615, -5.964101615}
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