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Simplifying (5x + 4yi)(5 + -4i) = (5x + -4y)(5 + -4i) Reorder the terms: (4iy + 5x)(5 + -4i) = (5x + -4y)(5 + -4i) Multiply (4iy + 5x) * (5 + -4i) (4iy * (5 + -4i) + 5x * (5 + -4i)) = (5x + -4y)(5 + -4i) ((5 * 4iy + -4i * 4iy) + 5x * (5 + -4i)) = (5x + -4y)(5 + -4i) ((20iy + -16i2y) + 5x * (5 + -4i)) = (5x + -4y)(5 + -4i) (20iy + -16i2y + (5 * 5x + -4i * 5x)) = (5x + -4y)(5 + -4i) Reorder the terms: (20iy + -16i2y + (-20ix + 25x)) = (5x + -4y)(5 + -4i) (20iy + -16i2y + (-20ix + 25x)) = (5x + -4y)(5 + -4i) Reorder the terms: (-20ix + 20iy + -16i2y + 25x) = (5x + -4y)(5 + -4i) (-20ix + 20iy + -16i2y + 25x) = (5x + -4y)(5 + -4i) Multiply (5x + -4y) * (5 + -4i) -20ix + 20iy + -16i2y + 25x = (5x * (5 + -4i) + -4y * (5 + -4i)) -20ix + 20iy + -16i2y + 25x = ((5 * 5x + -4i * 5x) + -4y * (5 + -4i)) Reorder the terms: -20ix + 20iy + -16i2y + 25x = ((-20ix + 25x) + -4y * (5 + -4i)) -20ix + 20iy + -16i2y + 25x = ((-20ix + 25x) + -4y * (5 + -4i)) -20ix + 20iy + -16i2y + 25x = (-20ix + 25x + (5 * -4y + -4i * -4y)) Reorder the terms: -20ix + 20iy + -16i2y + 25x = (-20ix + 25x + (16iy + -20y)) -20ix + 20iy + -16i2y + 25x = (-20ix + 25x + (16iy + -20y)) Reorder the terms: -20ix + 20iy + -16i2y + 25x = (-20ix + 16iy + 25x + -20y) -20ix + 20iy + -16i2y + 25x = (-20ix + 16iy + 25x + -20y) Add '20ix' to each side of the equation. -20ix + 20iy + -16i2y + 20ix + 25x = -20ix + 16iy + 25x + 20ix + -20y Reorder the terms: -20ix + 20ix + 20iy + -16i2y + 25x = -20ix + 16iy + 25x + 20ix + -20y Combine like terms: -20ix + 20ix = 0 0 + 20iy + -16i2y + 25x = -20ix + 16iy + 25x + 20ix + -20y 20iy + -16i2y + 25x = -20ix + 16iy + 25x + 20ix + -20y Reorder the terms: 20iy + -16i2y + 25x = -20ix + 20ix + 16iy + 25x + -20y Combine like terms: -20ix + 20ix = 0 20iy + -16i2y + 25x = 0 + 16iy + 25x + -20y 20iy + -16i2y + 25x = 16iy + 25x + -20y Add '-25x' to each side of the equation. 20iy + -16i2y + 25x + -25x = 16iy + 25x + -25x + -20y Combine like terms: 25x + -25x = 0 20iy + -16i2y + 0 = 16iy + 25x + -25x + -20y 20iy + -16i2y = 16iy + 25x + -25x + -20y Combine like terms: 25x + -25x = 0 20iy + -16i2y = 16iy + 0 + -20y 20iy + -16i2y = 16iy + -20y Solving 20iy + -16i2y = 16iy + -20y Solving for variable 'i'. Reorder the terms: 20iy + -16iy + -16i2y + 20y = 16iy + -20y + -16iy + 20y Combine like terms: 20iy + -16iy = 4iy 4iy + -16i2y + 20y = 16iy + -20y + -16iy + 20y Reorder the terms: 4iy + -16i2y + 20y = 16iy + -16iy + -20y + 20y Combine like terms: 16iy + -16iy = 0 4iy + -16i2y + 20y = 0 + -20y + 20y 4iy + -16i2y + 20y = -20y + 20y Combine like terms: -20y + 20y = 0 4iy + -16i2y + 20y = 0 Factor out the Greatest Common Factor (GCF), '4y'. 4y(i + -4i2 + 5) = 0 Factor a trinomial. 4y((-1i + -1)(4i + -5)) = 0 Ignore the factor 4.Subproblem 1
Set the factor 'y' equal to zero and attempt to solve: Simplifying y = 0 Solving y = 0 Move all terms containing i to the left, all other terms to the right. Add '-1y' to each side of the equation. y + -1y = 0 + -1y Remove the zero: 0 = -1y Simplifying 0 = -1y 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 '(-1i + -1)' equal to zero and attempt to solve: Simplifying -1i + -1 = 0 Reorder the terms: -1 + -1i = 0 Solving -1 + -1i = 0 Move all terms containing i to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + -1i = 0 + 1 Combine like terms: -1 + 1 = 0 0 + -1i = 0 + 1 -1i = 0 + 1 Combine like terms: 0 + 1 = 1 -1i = 1 Divide each side by '-1'. i = -1 Simplifying i = -1Subproblem 3
Set the factor '(4i + -5)' equal to zero and attempt to solve: Simplifying 4i + -5 = 0 Reorder the terms: -5 + 4i = 0 Solving -5 + 4i = 0 Move all terms containing i to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + 4i = 0 + 5 Combine like terms: -5 + 5 = 0 0 + 4i = 0 + 5 4i = 0 + 5 Combine like terms: 0 + 5 = 5 4i = 5 Divide each side by '4'. i = 1.25 Simplifying i = 1.25Solution
i = {-1, 1.25}
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