Complex Roots of Quadratics (DP IB Applications & Interpretation (AI)): Revision Note

Complex roots of quadratics

When does a quadratic have complex roots?

• The quadratic equation where has complex roots if

  • 

    • the discriminant is negative

How do I solve a quadratic with complex roots?

• To solve a quadratic equation with complex roots,

  • either use the quadratic formula

    • e.g.

  • or complete the square

How are complex roots related to complex conjugates?

• The two complex roots to the quadratic equation where are complex conjugate pairs

  • i.e. if is a root, then is the other root

• This is not true if any of the coefficients , or are complex

How do I factorise a quadratic expression using complex roots?

• If a quadratic expression has a negative discriminant (), then it can factorised as follows:

  • set the expression equal to zero

    • 

  • solve this equation to find the complex roots

    • and

  • rewrite in the factorised form

    • You could expand inside each bracket

    •