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Simplifying (2i)(7i) + -5(4 + 5i) = 0 Remove parenthesis around (2i) 2i(7i) + -5(4 + 5i) = 0 Remove parenthesis around (7i) 2i * 7i + -5(4 + 5i) = 0 Reorder the terms for easier multiplication: 2 * 7i * i + -5(4 + 5i) = 0 Multiply 2 * 7 14i * i + -5(4 + 5i) = 0 Multiply i * i 14i2 + -5(4 + 5i) = 0 14i2 + (4 * -5 + 5i * -5) = 0 14i2 + (-20 + -25i) = 0 Reorder the terms: -20 + -25i + 14i2 = 0 Solving -20 + -25i + 14i2 = 0 Solving for variable 'i'. Begin completing the square. Divide all terms by 14 the coefficient of the squared term: Divide each side by '14'. -1.428571429 + -1.785714286i + i2 = 0 Move the constant term to the right: Add '1.428571429' to each side of the equation. -1.428571429 + -1.785714286i + 1.428571429 + i2 = 0 + 1.428571429 Reorder the terms: -1.428571429 + 1.428571429 + -1.785714286i + i2 = 0 + 1.428571429 Combine like terms: -1.428571429 + 1.428571429 = 0.000000000 0.000000000 + -1.785714286i + i2 = 0 + 1.428571429 -1.785714286i + i2 = 0 + 1.428571429 Combine like terms: 0 + 1.428571429 = 1.428571429 -1.785714286i + i2 = 1.428571429 The i term is -1.785714286i. Take half its coefficient (-0.892857143). Square it (0.7971938778) and add it to both sides. Add '0.7971938778' to each side of the equation. -1.785714286i + 0.7971938778 + i2 = 1.428571429 + 0.7971938778 Reorder the terms: 0.7971938778 + -1.785714286i + i2 = 1.428571429 + 0.7971938778 Combine like terms: 1.428571429 + 0.7971938778 = 2.2257653068 0.7971938778 + -1.785714286i + i2 = 2.2257653068 Factor a perfect square on the left side: (i + -0.892857143)(i + -0.892857143) = 2.2257653068 Calculate the square root of the right side: 1.491899898 Break this problem into two subproblems by setting (i + -0.892857143) equal to 1.491899898 and -1.491899898.Subproblem 1
i + -0.892857143 = 1.491899898 Simplifying i + -0.892857143 = 1.491899898 Reorder the terms: -0.892857143 + i = 1.491899898 Solving -0.892857143 + i = 1.491899898 Solving for variable 'i'. Move all terms containing i to the left, all other terms to the right. Add '0.892857143' to each side of the equation. -0.892857143 + 0.892857143 + i = 1.491899898 + 0.892857143 Combine like terms: -0.892857143 + 0.892857143 = 0.000000000 0.000000000 + i = 1.491899898 + 0.892857143 i = 1.491899898 + 0.892857143 Combine like terms: 1.491899898 + 0.892857143 = 2.384757041 i = 2.384757041 Simplifying i = 2.384757041Subproblem 2
i + -0.892857143 = -1.491899898 Simplifying i + -0.892857143 = -1.491899898 Reorder the terms: -0.892857143 + i = -1.491899898 Solving -0.892857143 + i = -1.491899898 Solving for variable 'i'. Move all terms containing i to the left, all other terms to the right. Add '0.892857143' to each side of the equation. -0.892857143 + 0.892857143 + i = -1.491899898 + 0.892857143 Combine like terms: -0.892857143 + 0.892857143 = 0.000000000 0.000000000 + i = -1.491899898 + 0.892857143 i = -1.491899898 + 0.892857143 Combine like terms: -1.491899898 + 0.892857143 = -0.599042755 i = -0.599042755 Simplifying i = -0.599042755Solution
The solution to the problem is based on the solutions from the subproblems. i = {2.384757041, -0.599042755}
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