Answer to Question #252555 in Algebra for moe

Question

A complex number z satisfies the equation $z^{2}=3+4i.$

Which of the following is true about $z$ ?

1. $(3+4i)^{2}=25+24i$
2. $z=\pm (2+i)$
3. $(3+4i)^{2}=9+40i$
4. $z=\pm (2+2i)$
5. $(3+4i)^{2}=-7+ 24i$

Answer 1

Z2=3+4i,Z2-3-4i=0,

using quadratic formula,

Z=(−b±√(b² − 4ac))/2a

where a=1,b=0,c=(-3-4i)

Z=(-0±√(0²-4x1x(-3-4i)))/(2x1)

Z=(0±√(12+16i))/2

Z=(±(4+2i))/2

Z=±(2+i). Hence 2 is correct.

(3+4i)2=(3+4i)(3+4i)

(3+4i)2=9+12i+12i+(4i)2

(3+4i)2=9+24i+16(√(-1))2

(3+4i)2=9+24i+16(-1)

(3+4i)2=9+24i-16

(3+4i)2=(9-16)+24i

(3+4i)2=-7+24i.

Hence 5 is correct.

Therefore 2 and 5 are correct