Equation (a)

z2+(1−5i)z+(2i−6)=0 z^2 + (1 - 5i)z + (2i - 6) = 0

D=b2−4ac=(1−5i)2−4(1)(2i−6) D = b^2 - 4ac = (1 - 5i)^2 - 4(1)(2i - 6)

(1−5i)2=1−10i+25i2=1−10i−25=−24−10i (1 - 5i)^2 = 1 - 10i + 25i^2 = 1 - 10i - 25 = -24 - 10i

−4(2i−6)=−8i+24 -4(2i - 6) = -8i + 24

D=(−24−10i)+(24−8i)=−18i D = (-24 - 10i) + (24 - 8i) = -18i

Equation (b)

z2−(3+4i)z+(7i−1)=0 z^2 - (3 + 4i)z + (7i - 1) = 0

D=b2−4ac=[−(3+4i)]2−4(7i−1) D = b^2 - 4ac = [-(3 + 4i)]^2 - 4(7i - 1)

(−(3+4i))2=(−(3+4i))⋅(−(3+4i))=(3+4i)2 (-(3 + 4i))^2 = (-(3 + 4i)) \cdot (-(3 + 4i)) = (3 + 4i)^2

(3+4i)2=9+24i+16i2=9+24i−16=−7+24i (3 + 4i)^2 = 9 + 24i + 16i^2 = 9 + 24i - 16 = -7 + 24i

−4(7i−1)=−28i+4 -4(7i - 1) = -28i + 4

D=(−7+24i)+(4−28i)=−3−4i D = (-7 + 24i) + (4 - 28i) = -3 - 4i

Equation (c)

2z2+(5+i)z+(2+2i)=0 2z^2 + (5 + i)z + (2 + 2i) = 0

D=b2−4ac=(5+i)2−4(2)(2+2i) D = b^2 - 4ac = (5 + i)^2 - 4(2)(2 + 2i)

(5+i)2=25+10i+i2=25+10i−1=24+10i (5 + i)^2 = 25 + 10i + i^2 = 25 + 10i - 1 = 24 + 10i

4⋅2⋅(2+2i)=16+16i 4 \cdot 2 \cdot (2 + 2i) = 16 + 16i

D=(24+10i)−(16+16i)=8−6i D = (24 + 10i) - (16 + 16i) = 8 - 6i

Final Table

Equation Discriminant D=b2−4acD = b^2 - 4ac
z2+(1−5i)z+2i−6=0z^2 + (1 - 5i)z + 2i - 6 = 0 −18i-18i
z2−(3+4i)z+7i−1=0z^2 - (3 + 4i)z + 7i - 1 = 0 −3−4i-3 - 4i
2z2+(5+i)z+2+2i=02z^2 + (5 + i)z + 2 + 2i = 0 8−6i8 - 6i