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Simplifying 6j2 + -2j + 5 + 3j2 + 4j + -6 = 0 Reorder the terms: 5 + -6 + -2j + 4j + 6j2 + 3j2 = 0 Combine like terms: 5 + -6 = -1 -1 + -2j + 4j + 6j2 + 3j2 = 0 Combine like terms: -2j + 4j = 2j -1 + 2j + 6j2 + 3j2 = 0 Combine like terms: 6j2 + 3j2 = 9j2 -1 + 2j + 9j2 = 0 Solving -1 + 2j + 9j2 = 0 Solving for variable 'j'. Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -0.1111111111 + 0.2222222222j + j2 = 0 Move the constant term to the right: Add '0.1111111111' to each side of the equation. -0.1111111111 + 0.2222222222j + 0.1111111111 + j2 = 0 + 0.1111111111 Reorder the terms: -0.1111111111 + 0.1111111111 + 0.2222222222j + j2 = 0 + 0.1111111111 Combine like terms: -0.1111111111 + 0.1111111111 = 0.0000000000 0.0000000000 + 0.2222222222j + j2 = 0 + 0.1111111111 0.2222222222j + j2 = 0 + 0.1111111111 Combine like terms: 0 + 0.1111111111 = 0.1111111111 0.2222222222j + j2 = 0.1111111111 The j term is 0.2222222222j. Take half its coefficient (0.1111111111). Square it (0.01234567901) and add it to both sides. Add '0.01234567901' to each side of the equation. 0.2222222222j + 0.01234567901 + j2 = 0.1111111111 + 0.01234567901 Reorder the terms: 0.01234567901 + 0.2222222222j + j2 = 0.1111111111 + 0.01234567901 Combine like terms: 0.1111111111 + 0.01234567901 = 0.12345679011 0.01234567901 + 0.2222222222j + j2 = 0.12345679011 Factor a perfect square on the left side: (j + 0.1111111111)(j + 0.1111111111) = 0.12345679011 Calculate the square root of the right side: 0.351364184 Break this problem into two subproblems by setting (j + 0.1111111111) equal to 0.351364184 and -0.351364184.Subproblem 1
j + 0.1111111111 = 0.351364184 Simplifying j + 0.1111111111 = 0.351364184 Reorder the terms: 0.1111111111 + j = 0.351364184 Solving 0.1111111111 + j = 0.351364184 Solving for variable 'j'. Move all terms containing j to the left, all other terms to the right. Add '-0.1111111111' to each side of the equation. 0.1111111111 + -0.1111111111 + j = 0.351364184 + -0.1111111111 Combine like terms: 0.1111111111 + -0.1111111111 = 0.0000000000 0.0000000000 + j = 0.351364184 + -0.1111111111 j = 0.351364184 + -0.1111111111 Combine like terms: 0.351364184 + -0.1111111111 = 0.2402530729 j = 0.2402530729 Simplifying j = 0.2402530729Subproblem 2
j + 0.1111111111 = -0.351364184 Simplifying j + 0.1111111111 = -0.351364184 Reorder the terms: 0.1111111111 + j = -0.351364184 Solving 0.1111111111 + j = -0.351364184 Solving for variable 'j'. Move all terms containing j to the left, all other terms to the right. Add '-0.1111111111' to each side of the equation. 0.1111111111 + -0.1111111111 + j = -0.351364184 + -0.1111111111 Combine like terms: 0.1111111111 + -0.1111111111 = 0.0000000000 0.0000000000 + j = -0.351364184 + -0.1111111111 j = -0.351364184 + -0.1111111111 Combine like terms: -0.351364184 + -0.1111111111 = -0.4624752951 j = -0.4624752951 Simplifying j = -0.4624752951Solution
The solution to the problem is based on the solutions from the subproblems. j = {0.2402530729, -0.4624752951}
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