Solution of quadratic equations with real coefficients and a complex variable.

Solve the quadratic equation.

1. z2+2z+5=0.

Solution.

D=22415=420=16

z1=2+162=2+1612=2+4i2=1+2i;

z2=2162=21612=24i2=12i.

Answer: z1=1+2i; z2=12i.

2. 4z22z+1=0.

Solution.

D=(2)2441=416=12

z1=2+128=2+1218=2+23i8=14+34i;

z2=2128=21218=223i8=1434i.

Answer: z1=14+34i; z2=1434i.

Solve the biquadratic equation

3. z4+18z2+81=0.

Solution.

Let's make a variable substitution:

t=z2.

We get a quadratic equation:

t2+18t+81=0.

Let's solve it:

D=(18)24181=324324=0.

t1=18+02=9;

t2=1802=9.

Then let's make the inverse substitution:

t1=t2=z1,22=z3,42

9=z1,22

z1,2=±9=±91=±3i.

Answer: z1,2=z3,4=±3i.

Homework

Solve the biquadratic equation

1. z4+4z2+3=0.

Answer: z1,2=±i z3,4=±3i.

2. z4+9z2+20=0.

Answer: z1,2=±2i z3,4=±5i.

Tags: complex analysis, complex numbers, quadratic equations