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Simplifying 18k2 + 15k + -12 = -2 Reorder the terms: -12 + 15k + 18k2 = -2 Solving -12 + 15k + 18k2 = -2 Solving for variable 'k'. Reorder the terms: -12 + 2 + 15k + 18k2 = -2 + 2 Combine like terms: -12 + 2 = -10 -10 + 15k + 18k2 = -2 + 2 Combine like terms: -2 + 2 = 0 -10 + 15k + 18k2 = 0 Begin completing the square. Divide all terms by 18 the coefficient of the squared term: Divide each side by '18'. -0.5555555556 + 0.8333333333k + k2 = 0 Move the constant term to the right: Add '0.5555555556' to each side of the equation. -0.5555555556 + 0.8333333333k + 0.5555555556 + k2 = 0 + 0.5555555556 Reorder the terms: -0.5555555556 + 0.5555555556 + 0.8333333333k + k2 = 0 + 0.5555555556 Combine like terms: -0.5555555556 + 0.5555555556 = 0.0000000000 0.0000000000 + 0.8333333333k + k2 = 0 + 0.5555555556 0.8333333333k + k2 = 0 + 0.5555555556 Combine like terms: 0 + 0.5555555556 = 0.5555555556 0.8333333333k + k2 = 0.5555555556 The k term is 0.8333333333k. Take half its coefficient (0.4166666667). Square it (0.1736111111) and add it to both sides. Add '0.1736111111' to each side of the equation. 0.8333333333k + 0.1736111111 + k2 = 0.5555555556 + 0.1736111111 Reorder the terms: 0.1736111111 + 0.8333333333k + k2 = 0.5555555556 + 0.1736111111 Combine like terms: 0.5555555556 + 0.1736111111 = 0.7291666667 0.1736111111 + 0.8333333333k + k2 = 0.7291666667 Factor a perfect square on the left side: (k + 0.4166666667)(k + 0.4166666667) = 0.7291666667 Calculate the square root of the right side: 0.853912564 Break this problem into two subproblems by setting (k + 0.4166666667) equal to 0.853912564 and -0.853912564.Subproblem 1
k + 0.4166666667 = 0.853912564 Simplifying k + 0.4166666667 = 0.853912564 Reorder the terms: 0.4166666667 + k = 0.853912564 Solving 0.4166666667 + k = 0.853912564 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.4166666667' to each side of the equation. 0.4166666667 + -0.4166666667 + k = 0.853912564 + -0.4166666667 Combine like terms: 0.4166666667 + -0.4166666667 = 0.0000000000 0.0000000000 + k = 0.853912564 + -0.4166666667 k = 0.853912564 + -0.4166666667 Combine like terms: 0.853912564 + -0.4166666667 = 0.4372458973 k = 0.4372458973 Simplifying k = 0.4372458973Subproblem 2
k + 0.4166666667 = -0.853912564 Simplifying k + 0.4166666667 = -0.853912564 Reorder the terms: 0.4166666667 + k = -0.853912564 Solving 0.4166666667 + k = -0.853912564 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.4166666667' to each side of the equation. 0.4166666667 + -0.4166666667 + k = -0.853912564 + -0.4166666667 Combine like terms: 0.4166666667 + -0.4166666667 = 0.0000000000 0.0000000000 + k = -0.853912564 + -0.4166666667 k = -0.853912564 + -0.4166666667 Combine like terms: -0.853912564 + -0.4166666667 = -1.2705792307 k = -1.2705792307 Simplifying k = -1.2705792307Solution
The solution to the problem is based on the solutions from the subproblems. k = {0.4372458973, -1.2705792307}
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