Question

Given that z_(1)=x+5i and z_(2)=-5+9i where xinR, determine a simplified expression for (z_(1))/(z_(2)) in Cartesian form. (z_(1))/(z_(1))

Answer 1

`\[ (z_1)/(z_2) = ((x+5i))/((-5+9i)) * ((-5-9i))/((-5-9i)) = (-5x-25i+45)/(106) \]`

To simplify
`\((z_1)/(z_2)\)`in Cartesian form, multiply both the numerator and denominator by the conjugate of the denominator
`\((-5-9i)\)`. After simplification, the expression becomes
`\((-5x - 25i + 45)/(106)\)`.

Step-by-step explanation:

The expression
`\((z_1)/(z_2)\)`involves dividing complex numbers, which can be simplified by multiplying the numerator and denominator by the conjugate of the denominator. The conjugate of
`\((-5+9i)\) is \((-5-9i)\)`. Multiplying by the conjugate eliminates the imaginary part in the denominator.

By performing the multiplication and simplification, the expression becomes
`\((-5x-25i+45)/(106)\)`. The real part is
`\(-5x + (45)/(106)\)`, and the imaginary part is
`\(-(25)/(106)\)`. This result represents the Cartesian form of the quotient of the given complex numbers.

In summary, the simplified expression is obtained by manipulating the expression to eliminate the imaginary part from the denominator, resulting in a more manageable form for further calculations or analysis.