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Simplifying j + 1 = 4(j + -2) * 4j Reorder the terms: 1 + j = 4(j + -2) * 4j Reorder the terms: 1 + j = 4(-2 + j) * 4j Reorder the terms for easier multiplication: 1 + j = 4 * 4j(-2 + j) Multiply 4 * 4 1 + j = 16j(-2 + j) 1 + j = (-2 * 16j + j * 16j) 1 + j = (-32j + 16j2) Solving 1 + j = -32j + 16j2 Solving for variable 'j'. Combine like terms: j + 32j = 33j 1 + 33j + -16j2 = -32j + 16j2 + 32j + -16j2 Reorder the terms: 1 + 33j + -16j2 = -32j + 32j + 16j2 + -16j2 Combine like terms: -32j + 32j = 0 1 + 33j + -16j2 = 0 + 16j2 + -16j2 1 + 33j + -16j2 = 16j2 + -16j2 Combine like terms: 16j2 + -16j2 = 0 1 + 33j + -16j2 = 0 Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. -0.0625 + -2.0625j + j2 = 0 Move the constant term to the right: Add '0.0625' to each side of the equation. -0.0625 + -2.0625j + 0.0625 + j2 = 0 + 0.0625 Reorder the terms: -0.0625 + 0.0625 + -2.0625j + j2 = 0 + 0.0625 Combine like terms: -0.0625 + 0.0625 = 0.0000 0.0000 + -2.0625j + j2 = 0 + 0.0625 -2.0625j + j2 = 0 + 0.0625 Combine like terms: 0 + 0.0625 = 0.0625 -2.0625j + j2 = 0.0625 The j term is -2.0625j. Take half its coefficient (-1.03125). Square it (1.063476563) and add it to both sides. Add '1.063476563' to each side of the equation. -2.0625j + 1.063476563 + j2 = 0.0625 + 1.063476563 Reorder the terms: 1.063476563 + -2.0625j + j2 = 0.0625 + 1.063476563 Combine like terms: 0.0625 + 1.063476563 = 1.125976563 1.063476563 + -2.0625j + j2 = 1.125976563 Factor a perfect square on the left side: (j + -1.03125)(j + -1.03125) = 1.125976563 Calculate the square root of the right side: 1.061120428 Break this problem into two subproblems by setting (j + -1.03125) equal to 1.061120428 and -1.061120428.Subproblem 1
j + -1.03125 = 1.061120428 Simplifying j + -1.03125 = 1.061120428 Reorder the terms: -1.03125 + j = 1.061120428 Solving -1.03125 + j = 1.061120428 Solving for variable 'j'. Move all terms containing j to the left, all other terms to the right. Add '1.03125' to each side of the equation. -1.03125 + 1.03125 + j = 1.061120428 + 1.03125 Combine like terms: -1.03125 + 1.03125 = 0.00000 0.00000 + j = 1.061120428 + 1.03125 j = 1.061120428 + 1.03125 Combine like terms: 1.061120428 + 1.03125 = 2.092370428 j = 2.092370428 Simplifying j = 2.092370428Subproblem 2
j + -1.03125 = -1.061120428 Simplifying j + -1.03125 = -1.061120428 Reorder the terms: -1.03125 + j = -1.061120428 Solving -1.03125 + j = -1.061120428 Solving for variable 'j'. Move all terms containing j to the left, all other terms to the right. Add '1.03125' to each side of the equation. -1.03125 + 1.03125 + j = -1.061120428 + 1.03125 Combine like terms: -1.03125 + 1.03125 = 0.00000 0.00000 + j = -1.061120428 + 1.03125 j = -1.061120428 + 1.03125 Combine like terms: -1.061120428 + 1.03125 = -0.029870428 j = -0.029870428 Simplifying j = -0.029870428Solution
The solution to the problem is based on the solutions from the subproblems. j = {2.092370428, -0.029870428}
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