Divide the following complex numbers. (9-i)/(-4+5i)

Write Problem: Write down the problem to solve. We need to divide the complex number (9-i) by (-4+5i).Multiply by Conjugate: Multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of (-4+5i) is (-4-5i). We multiply both the numerator and the denominator by this conjugate to remove the imaginary part from the denominator. ((9-i) * (-4-5i)) / ((-4+5i) * (-4-5i))Multiply Numerator: Perform the multiplication in the numerator. (9-i) * (-4-5i) = 9*(-4) + 9*(-5i) - i*(-4) - i*(-5i) = -36 - 45i + 4i + 5 = -31 - 41iMultiply Denominator: Perform the multiplication in the denominator. (-4+5i) * (-4-5i) = (-4)*(-4) + (-4)*(-5i) + 5i*(-4) + 5i*(-5i) = 16 + 20i - 20i - 25i^2 Since i^2 = -1, we have: = 16 - 25(-1) = 16 + 25 = 41Divide Results: Divide the results from Step 3 by the result from Step 4. (-31 - 41i) / 41 = -31/41 - (41i/41) = -31/41 - i

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Algebra 2

Add, subtract, multiply, and divide polynomials

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Q. Divide the following complex numbers.

\newline

\frac{9-i}{-4+5 i}

Write Problem: Write down the problem to solve. $\newline$ We need to divide the complex number $(9-i)$ by $(-4+5i)$ .
Multiply by Conjugate: Multiply the numerator and the denominator by the conjugate of the denominator. $\newline$ The conjugate of $(-4+5i)$ is $(-4-5i)$ . We multiply both the numerator and the denominator by this conjugate to remove the imaginary part from the denominator. $\newline$ $\frac{(9-i) \cdot (-4-5i)}{(-4+5i) \cdot (-4-5i)}$
Multiply Numerator: Perform the multiplication in the numerator. $\newline$ $(9-i) \times (-4-5i) = 9\times(-4) + 9\times(-5i) - i\times(-4) - i\times(-5i)$ $\newline$ $= -36 - 45i + 4i + 5$ $\newline$ $= -31 - 41i$
Multiply Denominator: Perform the multiplication in the denominator. $\newline$ $(-4+5i) \times (-4-5i) = (-4)\times(-4) + (-4)\times(-5i) + 5i\times(-4) + 5i\times(-5i)(\newline$ = 16 + 20i - 20i - 25i^2(\newline\)Since \$i^2 = -1\), we have:(\newline\)= $16$ - $25$($-1$)(\newline\)= $16$ + $25$(\newline\)= $41$
Divide Results: Divide the results from Step $3$ by the result from Step $4$.$\newline$$(-31 - 41i) / 41$$\newline$$= -\frac{31}{41} - (\frac{41i}{41})$$\newline$$= -\frac{31}{41} - i$