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Simplifying 36k2 + -99k + -30 = 0 Reorder the terms: -30 + -99k + 36k2 = 0 Solving -30 + -99k + 36k2 = 0 Solving for variable 'k'. Factor out the Greatest Common Factor (GCF), '3'. 3(-10 + -33k + 12k2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-10 + -33k + 12k2)' equal to zero and attempt to solve: Simplifying -10 + -33k + 12k2 = 0 Solving -10 + -33k + 12k2 = 0 Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. -0.8333333333 + -2.75k + k2 = 0 Move the constant term to the right: Add '0.8333333333' to each side of the equation. -0.8333333333 + -2.75k + 0.8333333333 + k2 = 0 + 0.8333333333 Reorder the terms: -0.8333333333 + 0.8333333333 + -2.75k + k2 = 0 + 0.8333333333 Combine like terms: -0.8333333333 + 0.8333333333 = 0.0000000000 0.0000000000 + -2.75k + k2 = 0 + 0.8333333333 -2.75k + k2 = 0 + 0.8333333333 Combine like terms: 0 + 0.8333333333 = 0.8333333333 -2.75k + k2 = 0.8333333333 The k term is -2.75k. Take half its coefficient (-1.375). Square it (1.890625) and add it to both sides. Add '1.890625' to each side of the equation. -2.75k + 1.890625 + k2 = 0.8333333333 + 1.890625 Reorder the terms: 1.890625 + -2.75k + k2 = 0.8333333333 + 1.890625 Combine like terms: 0.8333333333 + 1.890625 = 2.7239583333 1.890625 + -2.75k + k2 = 2.7239583333 Factor a perfect square on the left side: (k + -1.375)(k + -1.375) = 2.7239583333 Calculate the square root of the right side: 1.65044186 Break this problem into two subproblems by setting (k + -1.375) equal to 1.65044186 and -1.65044186.Subproblem 1
k + -1.375 = 1.65044186 Simplifying k + -1.375 = 1.65044186 Reorder the terms: -1.375 + k = 1.65044186 Solving -1.375 + k = 1.65044186 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '1.375' to each side of the equation. -1.375 + 1.375 + k = 1.65044186 + 1.375 Combine like terms: -1.375 + 1.375 = 0.000 0.000 + k = 1.65044186 + 1.375 k = 1.65044186 + 1.375 Combine like terms: 1.65044186 + 1.375 = 3.02544186 k = 3.02544186 Simplifying k = 3.02544186Subproblem 2
k + -1.375 = -1.65044186 Simplifying k + -1.375 = -1.65044186 Reorder the terms: -1.375 + k = -1.65044186 Solving -1.375 + k = -1.65044186 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '1.375' to each side of the equation. -1.375 + 1.375 + k = -1.65044186 + 1.375 Combine like terms: -1.375 + 1.375 = 0.000 0.000 + k = -1.65044186 + 1.375 k = -1.65044186 + 1.375 Combine like terms: -1.65044186 + 1.375 = -0.27544186 k = -0.27544186 Simplifying k = -0.27544186Solution
The solution to the problem is based on the solutions from the subproblems. k = {3.02544186, -0.27544186}Solution
k = {3.02544186, -0.27544186}
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