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Simplify the complex number expression z = (5)/(2-i) - (3+2i) and express it in the form x+iy.

Mathematics

ScanSolve AI Answer

Answer

-1-i

Step-by-step solution

4 steps

❌ Incorrect. Your calculation for zz is incorrect.

  1. Identify the error in the student's work.

    The main error is in the first step of your solution. You incorrectly transformed the expression 52−i−(3+2i)\frac{5}{2-i} - (3+2i) into 5−(3+2i)(2−i)5 - (3+2i)(2-i). The term 52−i\frac{5}{2-i} must be simplified by multiplying the numerator and denominator by the conjugate of the denominator, not by moving the denominator to multiply the second term.

  2. Correctly simplify the first term.

    52−i=52−i×2+i2+i\frac{5}{2-i} = \frac{5}{2-i} \times \frac{2+i}{2+i} =5(2+i)22−i2= \frac{5(2+i)}{2^2 - i^2} =10+5i4−(−1)= \frac{10+5i}{4 - (-1)} =10+5i5= \frac{10+5i}{5} =2+i= 2+i

  3. Substitute the simplified term back into the expression for zz.

    z=(2+i)−(3+2i)z = (2+i) - (3+2i)

  4. Perform the subtraction.

    z=2+i−3−2iz = 2+i-3-2i z=(2−3)+(1−2)iz = (2-3) + (1-2)i z=−1−iz = -1-i

    The correct value for zz in the form x+iyx+iy is −1−i\boxed{-1-i}.

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