Question - Solving a Complex Fraction to Standard Form
Solution:
The provided expression is a complex fraction:\[ \frac{8 - i}{3 - 2i} \]To write this in the standard form $$ a + bi $$, where $$ a $$ and $$ b $$ are real numbers, you must remove the imaginary part from the denominator. You can do this by multiplying both the numerator and the denominator by the complex conjugate of the denominator. The complex conjugate of $$3 - 2i$$ is $$3 + 2i$$. Multiplying the numerator and the denominator by $$3 + 2i$$ gives us:\[ \frac{8 - i}{3 - 2i} \times \frac{3 + 2i}{3 + 2i} \]This multiplication will give us:\[ \frac{(8 - i)(3 + 2i)}{(3 - 2i)(3 + 2i)} \]Expand the numerator and the denominator:\[ \frac{24 + 16i - 3i - 2i^2}{9 - 6i + 6i - 4i^2} \]Since $$i^2 = -1$$, this simplifies to:\[ \frac{24 + 13i - 2(-1)}{9 - 4(-1)} \]\[ \frac{24 + 13i + 2}{9 + 4} \]\[ \frac{26 + 13i}{13} \]\[ 2 + i \]So in the form $$ a + bi $$, $$ a = 2 $$ and $$ b = 1 $$. Therefore, the value of $$ a $$ is 2, which corresponds to option A.
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