+24 Fill in the blanks with real numbers.
\(11-7i/2+i\) = __ + __i
The answer is NOT 3 + 5i
+400 To simplify the expression (11 - 7i)/(2 + i), we can use the method of multiplying both numerator and denominator by the conjugate of the denominator.
The conjugate of the denominator, 2 + i, is 2 - i. Therefore, we have:
(11 - 7i)/(2 + i) = (11 - 7i)/(2 + i) * (2 - i) / (2 - i)
Expanding the numerator and denominator:
= [(11 - 7i) * (2 - i)] / [(2 + i) * (2 - i)]
= [(11 * 2) + (-7 * 2i) + (11 * -1i) + (-7 * -1 * i^2)] / [(2 * 2) + (2 * -1i) + (1 * -1i) + (1 * -1i^2)]
= [(22 - 14i - 11i + 7 i^2)] / [(4 - 2i - 1i - 1 * i^2)]
= [(22 - 14i - 11i - 7)] / [(4 - 2i - 1i + 1)]
= [15 - 25i] / [5 - 3i]
= (15 - 25i) / 5(1 - 0.6i)
Dividing both numerator and denominator by 5:
= (3 - 5i) / (1 - 0.6i)
Now, multiplying both numerator and denominator by the conjugate of the new denominator (1 + 0.6i):
= (3 - 5i) * (1 + 0.6i) / (1 - 0.6i) * (1 + 0.6i)
Expanding again:
= [(3 * 1) + (-5 * 0.6i) + (3 * 0.6i) + (-5 * 0.6 * i^2)] / [(1 * 1) + (1 * 0.6i) + (-0.6 * 1i) + (-0.6 * 0.6 * i^2)]
= [3 - 3i + 1.8i + 3] / [1 + 0.6i - 0.6i - 0.36]
= [6 - 1.2i] / [0.64]
= (6 - 1.2i) / (0.8 * 0.8)
= (6 - 1.2i) / 0.64
= (15 - 3i) / 4
= 3 - \frac{3}{4}i
Therefore, the simplified form of the complex number is 3 - 3/4*i, which you can write as
3 + (-3/4) * i
Yay for complex numbers!
