Write in Standard Form (4-i)^2-(1+2i)^2

Problem

(4−i)2−(1+2*i)2

Solution

1. Expand the first squared binomial (4−i)2 using the identity (a−b)2=a2−2*a*b+b2

(4−i)2=4−2*(4)*(i)+i2

16−8*i+i2

2. Expand the second squared binomial (1+2*i)2 using the identity (a+b)2=a2+2*a*b+b2

(1+2*i)2=1+2*(1)*(2*i)+(2*i)2

1+4*i+4*i2

3. Substitute i2=−1 into both expanded expressions to simplify them.

(4−i)2=16−8*i−1=15−8*i

(1+2*i)2=1+4*i−4=−3+4*i

4. Subtract the second simplified expression from the first.

(15−8*i)−(−3+4*i)

5. Distribute the negative sign and combine the real and imaginary parts.

15−8*i+3−4*i

(15+3)+(−8*i−4*i)

18−12*i

Final Answer

(4−i)2−(1+2*i)2=18−12*i

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