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Simplifying -6i2 + 5i + 15 = 0 Reorder the terms: 15 + 5i + -6i2 = 0 Solving 15 + 5i + -6i2 = 0 Solving for variable 'i'. Begin completing the square. Divide all terms by -6 the coefficient of the squared term: Divide each side by '-6'. -2.5 + -0.8333333333i + i2 = 0 Move the constant term to the right: Add '2.5' to each side of the equation. -2.5 + -0.8333333333i + 2.5 + i2 = 0 + 2.5 Reorder the terms: -2.5 + 2.5 + -0.8333333333i + i2 = 0 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + -0.8333333333i + i2 = 0 + 2.5 -0.8333333333i + i2 = 0 + 2.5 Combine like terms: 0 + 2.5 = 2.5 -0.8333333333i + i2 = 2.5 The i term is -0.8333333333i. 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. -0.8333333333i + 0.1736111111 + i2 = 2.5 + 0.1736111111 Reorder the terms: 0.1736111111 + -0.8333333333i + i2 = 2.5 + 0.1736111111 Combine like terms: 2.5 + 0.1736111111 = 2.6736111111 0.1736111111 + -0.8333333333i + i2 = 2.6736111111 Factor a perfect square on the left side: (i + -0.4166666667)(i + -0.4166666667) = 2.6736111111 Calculate the square root of the right side: 1.635118073 Break this problem into two subproblems by setting (i + -0.4166666667) equal to 1.635118073 and -1.635118073.Subproblem 1
i + -0.4166666667 = 1.635118073 Simplifying i + -0.4166666667 = 1.635118073 Reorder the terms: -0.4166666667 + i = 1.635118073 Solving -0.4166666667 + i = 1.635118073 Solving for variable 'i'. Move all terms containing i to the left, all other terms to the right. Add '0.4166666667' to each side of the equation. -0.4166666667 + 0.4166666667 + i = 1.635118073 + 0.4166666667 Combine like terms: -0.4166666667 + 0.4166666667 = 0.0000000000 0.0000000000 + i = 1.635118073 + 0.4166666667 i = 1.635118073 + 0.4166666667 Combine like terms: 1.635118073 + 0.4166666667 = 2.0517847397 i = 2.0517847397 Simplifying i = 2.0517847397Subproblem 2
i + -0.4166666667 = -1.635118073 Simplifying i + -0.4166666667 = -1.635118073 Reorder the terms: -0.4166666667 + i = -1.635118073 Solving -0.4166666667 + i = -1.635118073 Solving for variable 'i'. Move all terms containing i to the left, all other terms to the right. Add '0.4166666667' to each side of the equation. -0.4166666667 + 0.4166666667 + i = -1.635118073 + 0.4166666667 Combine like terms: -0.4166666667 + 0.4166666667 = 0.0000000000 0.0000000000 + i = -1.635118073 + 0.4166666667 i = -1.635118073 + 0.4166666667 Combine like terms: -1.635118073 + 0.4166666667 = -1.2184514063 i = -1.2184514063 Simplifying i = -1.2184514063Solution
The solution to the problem is based on the solutions from the subproblems. i = {2.0517847397, -1.2184514063}
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