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Express as a complex number in simplest a+bi form:

(24-13 i)/(2+i)
Answer:

Express as a complex number in simplest a+bi form: $\newline$ $\frac{24-13 i}{2+i}$ $\newline$ Answer:

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Write a quadratic function from its zeros

Full solution

Q. Express as a complex number in simplest a+bi form:

\newline

\frac{24-13 i}{2+i}

\newline

Answer:

Multiply by Conjugate: Multiply the numerator and denominator by the conjugate of the denominator to remove the imaginary unit from the denominator. $\newline$ The conjugate of $2+i$ is $2-i$ . $\newline$ So, we multiply $(24-13i)/(2+i)$ by $(2-i)/(2-i)$ to rationalize the denominator.
Numerator Multiplication: Perform the multiplication in the numerator. $\newline$ Multiply $(24-13i)$ by $(2-i)$ . $\newline$ Using the distributive property (FOIL method): $\newline$ $24\cdot 2 + 24\cdot (-i) - 13i\cdot 2 - 13i\cdot (-i)$ $\newline$ = $48 - 24i - 26i + 13i^2$ $\newline$ Since $i^2 = -1$ , replace $13i^2$ with $-13$ . $\newline$ = $48 - 24i - 26i - 13$ $\newline$ = $48 - 13 - (24i + 26i)$ $\newline$ = $35 - 50i$
Denominator Multiplication: Perform the multiplication in the denominator. $\newline$ Multiply $(2+i)$ by $(2-i)$ . $\newline$ Using the difference of squares: $\newline$ $2\times 2 - i\times 2 + i\times 2 - i^2$ $\newline$ $= 4 - i^2$ $\newline$ Since $i^2 = -1$ , replace $-i^2$ with $1$ . $\newline$ $= 4 + 1$ $\newline$ $= 5$
Final Complex Number: Divide the results from Step $2$ by the result from Step $3$ to get the complex number in $a+bi$ form. $\newline$ $f(x) = \frac{35 - 50i}{5}$ $\newline$ Divide both the real part and the imaginary part by $5$ . $\newline$ $= \frac{35}{5} - \frac{50i}{5}$ $\newline$ $= 7 - 10i$

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