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Simplifying (4 + 6.9i)(-2 + 5i) = 0 Multiply (4 + 6.9i) * (-2 + 5i) (4(-2 + 5i) + 6.9i * (-2 + 5i)) = 0 ((-2 * 4 + 5i * 4) + 6.9i * (-2 + 5i)) = 0 ((-8 + 20i) + 6.9i * (-2 + 5i)) = 0 (-8 + 20i + (-2 * 6.9i + 5i * 6.9i)) = 0 (-8 + 20i + (-13.8i + 34.5i2)) = 0 Combine like terms: 20i + -13.8i = 6.2i (-8 + 6.2i + 34.5i2) = 0 Solving -8 + 6.2i + 34.5i2 = 0 Solving for variable 'i'. Begin completing the square. Divide all terms by 34.5 the coefficient of the squared term: Divide each side by '34.5'. -0.231884058 + 0.1797101449i + i2 = 0 Move the constant term to the right: Add '0.231884058' to each side of the equation. -0.231884058 + 0.1797101449i + 0.231884058 + i2 = 0 + 0.231884058 Reorder the terms: -0.231884058 + 0.231884058 + 0.1797101449i + i2 = 0 + 0.231884058 Combine like terms: -0.231884058 + 0.231884058 = 0.000000000 0.000000000 + 0.1797101449i + i2 = 0 + 0.231884058 0.1797101449i + i2 = 0 + 0.231884058 Combine like terms: 0 + 0.231884058 = 0.231884058 0.1797101449i + i2 = 0.231884058 The i term is 0.1797101449i. Take half its coefficient (0.08985507245). Square it (0.008073934045) and add it to both sides. Add '0.008073934045' to each side of the equation. 0.1797101449i + 0.008073934045 + i2 = 0.231884058 + 0.008073934045 Reorder the terms: 0.008073934045 + 0.1797101449i + i2 = 0.231884058 + 0.008073934045 Combine like terms: 0.231884058 + 0.008073934045 = 0.239957992045 0.008073934045 + 0.1797101449i + i2 = 0.239957992045 Factor a perfect square on the left side: (i + 0.08985507245)(i + 0.08985507245) = 0.239957992045 Calculate the square root of the right side: 0.489855072 Break this problem into two subproblems by setting (i + 0.08985507245) equal to 0.489855072 and -0.489855072.Subproblem 1
i + 0.08985507245 = 0.489855072 Simplifying i + 0.08985507245 = 0.489855072 Reorder the terms: 0.08985507245 + i = 0.489855072 Solving 0.08985507245 + i = 0.489855072 Solving for variable 'i'. Move all terms containing i to the left, all other terms to the right. Add '-0.08985507245' to each side of the equation. 0.08985507245 + -0.08985507245 + i = 0.489855072 + -0.08985507245 Combine like terms: 0.08985507245 + -0.08985507245 = 0.00000000000 0.00000000000 + i = 0.489855072 + -0.08985507245 i = 0.489855072 + -0.08985507245 Combine like terms: 0.489855072 + -0.08985507245 = 0.39999999955 i = 0.39999999955 Simplifying i = 0.39999999955Subproblem 2
i + 0.08985507245 = -0.489855072 Simplifying i + 0.08985507245 = -0.489855072 Reorder the terms: 0.08985507245 + i = -0.489855072 Solving 0.08985507245 + i = -0.489855072 Solving for variable 'i'. Move all terms containing i to the left, all other terms to the right. Add '-0.08985507245' to each side of the equation. 0.08985507245 + -0.08985507245 + i = -0.489855072 + -0.08985507245 Combine like terms: 0.08985507245 + -0.08985507245 = 0.00000000000 0.00000000000 + i = -0.489855072 + -0.08985507245 i = -0.489855072 + -0.08985507245 Combine like terms: -0.489855072 + -0.08985507245 = -0.57971014445 i = -0.57971014445 Simplifying i = -0.57971014445Solution
The solution to the problem is based on the solutions from the subproblems. i = {0.39999999955, -0.57971014445}
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