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Simplifying (2 + -7i) * x(9 + 5i) = x Reorder the terms for easier multiplication: x(2 + -7i)(9 + 5i) = x Multiply (2 + -7i) * (9 + 5i) x(2(9 + 5i) + -7i * (9 + 5i)) = x x((9 * 2 + 5i * 2) + -7i * (9 + 5i)) = x x((18 + 10i) + -7i * (9 + 5i)) = x x(18 + 10i + (9 * -7i + 5i * -7i)) = x x(18 + 10i + (-63i + -35i2)) = x Combine like terms: 10i + -63i = -53i x(18 + -53i + -35i2) = x (18 * x + -53i * x + -35i2 * x) = x Reorder the terms: (-53ix + -35i2x + 18x) = x (-53ix + -35i2x + 18x) = x Solving -53ix + -35i2x + 18x = x Solving for variable 'i'. Combine like terms: 18x + -1x = 17x -53ix + -35i2x + 17x = x + -1x Combine like terms: x + -1x = 0 -53ix + -35i2x + 17x = 0 Factor out the Greatest Common Factor (GCF), 'x'. x(-53i + -35i2 + 17) = 0Subproblem 1
Set the factor 'x' equal to zero and attempt to solve: Simplifying x = 0 Solving x = 0 Move all terms containing i to the left, all other terms to the right. Add '-1x' to each side of the equation. x + -1x = 0 + -1x Remove the zero: 0 = -1x Simplifying 0 = -1x The solution to this equation could not be determined. This subproblem is being ignored because a solution could not be determined.Subproblem 2
Set the factor '(-53i + -35i2 + 17)' equal to zero and attempt to solve: Simplifying -53i + -35i2 + 17 = 0 Reorder the terms: 17 + -53i + -35i2 = 0 Solving 17 + -53i + -35i2 = 0 Begin completing the square. Divide all terms by -35 the coefficient of the squared term: Divide each side by '-35'. -0.4857142857 + 1.514285714i + i2 = 0 Move the constant term to the right: Add '0.4857142857' to each side of the equation. -0.4857142857 + 1.514285714i + 0.4857142857 + i2 = 0 + 0.4857142857 Reorder the terms: -0.4857142857 + 0.4857142857 + 1.514285714i + i2 = 0 + 0.4857142857 Combine like terms: -0.4857142857 + 0.4857142857 = 0.0000000000 0.0000000000 + 1.514285714i + i2 = 0 + 0.4857142857 1.514285714i + i2 = 0 + 0.4857142857 Combine like terms: 0 + 0.4857142857 = 0.4857142857 1.514285714i + i2 = 0.4857142857 The i term is 1.514285714i. Take half its coefficient (0.757142857). Square it (0.5732653059) and add it to both sides. Add '0.5732653059' to each side of the equation. 1.514285714i + 0.5732653059 + i2 = 0.4857142857 + 0.5732653059 Reorder the terms: 0.5732653059 + 1.514285714i + i2 = 0.4857142857 + 0.5732653059 Combine like terms: 0.4857142857 + 0.5732653059 = 1.0589795916 0.5732653059 + 1.514285714i + i2 = 1.0589795916 Factor a perfect square on the left side: (i + 0.757142857)(i + 0.757142857) = 1.0589795916 Calculate the square root of the right side: 1.029067341 Break this problem into two subproblems by setting (i + 0.757142857) equal to 1.029067341 and -1.029067341.Subproblem 1
i + 0.757142857 = 1.029067341 Simplifying i + 0.757142857 = 1.029067341 Reorder the terms: 0.757142857 + i = 1.029067341 Solving 0.757142857 + i = 1.029067341 Solving for variable 'i'. Move all terms containing i to the left, all other terms to the right. Add '-0.757142857' to each side of the equation. 0.757142857 + -0.757142857 + i = 1.029067341 + -0.757142857 Combine like terms: 0.757142857 + -0.757142857 = 0.000000000 0.000000000 + i = 1.029067341 + -0.757142857 i = 1.029067341 + -0.757142857 Combine like terms: 1.029067341 + -0.757142857 = 0.271924484 i = 0.271924484 Simplifying i = 0.271924484Subproblem 2
i + 0.757142857 = -1.029067341 Simplifying i + 0.757142857 = -1.029067341 Reorder the terms: 0.757142857 + i = -1.029067341 Solving 0.757142857 + i = -1.029067341 Solving for variable 'i'. Move all terms containing i to the left, all other terms to the right. Add '-0.757142857' to each side of the equation. 0.757142857 + -0.757142857 + i = -1.029067341 + -0.757142857 Combine like terms: 0.757142857 + -0.757142857 = 0.000000000 0.000000000 + i = -1.029067341 + -0.757142857 i = -1.029067341 + -0.757142857 Combine like terms: -1.029067341 + -0.757142857 = -1.786210198 i = -1.786210198 Simplifying i = -1.786210198Solution
The solution to the problem is based on the solutions from the subproblems. i = {0.271924484, -1.786210198}Solution
i = {0.271924484, -1.786210198}
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