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Simplifying 5k2 + -125 + k = 0 Reorder the terms: -125 + k + 5k2 = 0 Solving -125 + k + 5k2 = 0 Solving for variable 'k'. Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -25 + 0.2k + k2 = 0 Move the constant term to the right: Add '25' to each side of the equation. -25 + 0.2k + 25 + k2 = 0 + 25 Reorder the terms: -25 + 25 + 0.2k + k2 = 0 + 25 Combine like terms: -25 + 25 = 0 0 + 0.2k + k2 = 0 + 25 0.2k + k2 = 0 + 25 Combine like terms: 0 + 25 = 25 0.2k + k2 = 25 The k term is k. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. 0.2k + 0.25 + k2 = 25 + 0.25 Reorder the terms: 0.25 + 0.2k + k2 = 25 + 0.25 Combine like terms: 25 + 0.25 = 25.25 0.25 + 0.2k + k2 = 25.25 Factor a perfect square on the left side: (k + 0.5)(k + 0.5) = 25.25 Calculate the square root of the right side: 5.024937811 Break this problem into two subproblems by setting (k + 0.5) equal to 5.024937811 and -5.024937811.Subproblem 1
k + 0.5 = 5.024937811 Simplifying k + 0.5 = 5.024937811 Reorder the terms: 0.5 + k = 5.024937811 Solving 0.5 + k = 5.024937811 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + k = 5.024937811 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + k = 5.024937811 + -0.5 k = 5.024937811 + -0.5 Combine like terms: 5.024937811 + -0.5 = 4.524937811 k = 4.524937811 Simplifying k = 4.524937811Subproblem 2
k + 0.5 = -5.024937811 Simplifying k + 0.5 = -5.024937811 Reorder the terms: 0.5 + k = -5.024937811 Solving 0.5 + k = -5.024937811 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + k = -5.024937811 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + k = -5.024937811 + -0.5 k = -5.024937811 + -0.5 Combine like terms: -5.024937811 + -0.5 = -5.524937811 k = -5.524937811 Simplifying k = -5.524937811Solution
The solution to the problem is based on the solutions from the subproblems. k = {4.524937811, -5.524937811}
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