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Simplifying -3i * 6i + -3(-7 + 6i) = 0 Reorder the terms for easier multiplication: -3 * 6i * i + -3(-7 + 6i) = 0 Multiply -3 * 6 -18i * i + -3(-7 + 6i) = 0 Multiply i * i -18i2 + -3(-7 + 6i) = 0 -18i2 + (-7 * -3 + 6i * -3) = 0 -18i2 + (21 + -18i) = 0 Reorder the terms: 21 + -18i + -18i2 = 0 Solving 21 + -18i + -18i2 = 0 Solving for variable 'i'. Factor out the Greatest Common Factor (GCF), '3'. 3(7 + -6i + -6i2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(7 + -6i + -6i2)' equal to zero and attempt to solve: Simplifying 7 + -6i + -6i2 = 0 Solving 7 + -6i + -6i2 = 0 Begin completing the square. Divide all terms by -6 the coefficient of the squared term: Divide each side by '-6'. -1.166666667 + i + i2 = 0 Move the constant term to the right: Add '1.166666667' to each side of the equation. -1.166666667 + i + 1.166666667 + i2 = 0 + 1.166666667 Reorder the terms: -1.166666667 + 1.166666667 + i + i2 = 0 + 1.166666667 Combine like terms: -1.166666667 + 1.166666667 = 0.000000000 0.000000000 + i + i2 = 0 + 1.166666667 i + i2 = 0 + 1.166666667 Combine like terms: 0 + 1.166666667 = 1.166666667 i + i2 = 1.166666667 The i term is i. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. + 0.25 + i2 = 1.166666667 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + i2 = 1.166666667 + 0.25 Combine like terms: 1.166666667 + 0.25 = 1.416666667 1.25 + i2 = 1.416666667 Factor a perfect square on the left side: (i + 0.5)(i + 0.5) = 1.416666667 Calculate the square root of the right side: 1.190238072 Break this problem into two subproblems by setting (i + 0.5) equal to 1.190238072 and -1.190238072.Subproblem 1
i + 0.5 = 1.190238072 Simplifying i + 0.5 = 1.190238072 Reorder the terms: 0.5 + i = 1.190238072 Solving 0.5 + i = 1.190238072 Solving for variable 'i'. Move all terms containing i to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + i = 1.190238072 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + i = 1.190238072 + -0.5 i = 1.190238072 + -0.5 Combine like terms: 1.190238072 + -0.5 = 0.690238072 i = 0.690238072 Simplifying i = 0.690238072Subproblem 2
i + 0.5 = -1.190238072 Simplifying i + 0.5 = -1.190238072 Reorder the terms: 0.5 + i = -1.190238072 Solving 0.5 + i = -1.190238072 Solving for variable 'i'. Move all terms containing i to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + i = -1.190238072 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + i = -1.190238072 + -0.5 i = -1.190238072 + -0.5 Combine like terms: -1.190238072 + -0.5 = -1.690238072 i = -1.690238072 Simplifying i = -1.690238072Solution
The solution to the problem is based on the solutions from the subproblems. i = {0.690238072, -1.690238072}Solution
i = {0.690238072, -1.690238072}
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