Complex Numbers and Quadratic Equations
Complex Numbers and Quadratic Equations covers i^2=-1, modulus, argument, conjugate, polar form, roots of quadratics and the relation between roots and coefficients.
Core rules
- For z=a+bi, modulus |z|=sqrt(a^2+b^2) and conjugate is a-bi. z times conjugate = |z|^2.
- Division is done by multiplying numerator and denominator by the conjugate of the denominator.
- A quadratic ax^2+bx+c=0 has roots (-b±sqrt(D))/(2a), where D=b^2-4ac. D>0 gives real distinct roots, D=0 equal real roots, D<0 conjugate complex roots.
- Sum of roots=-b/a and product=c/a. For real coefficients, non-real roots occur in conjugate pairs.
Worked example
Solve x^2-4x+13=0. Here D=16-52=-36, so sqrt D=6i. The roots are (4±6i)/2=2±3i. Their sum is 4 and product is 2^2+3^2=13, matching -b/a and c/a.
Traps
- sqrt(-36) is 6i, not -6 or undefined.
- The conjugate of a product is the product of conjugates, but the argument of a product is the sum of arguments only up to 2π.
- In polar form, the angle must match the quadrant of a+bi.
- Do not use the conjugate pair rule when coefficients are not real.
- Check that the quadratic is written in standard form before reading a, b and c.