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

Complex Roots of Quadratics

What are complex roots?

• A quadratic equation can either have two real roots (zeros), a repeated real root or no real roots

  • This depends on the location of the graph of the quadratic with respect to the x-axis

• If a quadratic equation has no real roots we would previously have stated that it has no real solutions

  • The quadratic equation will have a negative discriminant

  • This means taking the square root of a negative number

• Complex numbers provide solutions for quadratic equations that have no real roots

How do we solve a quadratic equation when it has complex roots?

• If a quadratic equation takes the form ax² + bx + c = 0 it can be solved by either using the quadratic formula or completing the square

• If a quadratic equation takes the form ax² + b = 0 it can be solved by rearranging

• The property i = √-1 is used

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• If the coefficients of the quadratic are real then the complex roots will occur in complex conjugate pairs

  • If z = p + qi (q ≠ 0) is a root of a quadratic with real coefficients then z* = p - qi is also a root

• The real part of the solutions will have the same value as the x coordinate of the turning point on the graph of the quadratic

• When the coefficients of the quadratic equation are non-real, the solutions will not be complex conjugates

  • To solve these you can use the quadratic formula

How do we factorise a quadratic equation if it has complex roots?

• If we are given a quadratic equation in the form az² + bz + c = 0, where a, b, and c ∈ ℝ, a ≠ 0 we can use its complex roots to write it in factorised form

  • Use the quadratic formula to find the two roots, z  = p + qi and z* = p - qi

  • This means that z – (p + qi) and z – (p – qi) must both be factors of the quadratic equation

  • Therefore we can write az² + bz + c = a(z – (p + qi))( z – (p - qi))

  • This can be rearranged into the form a(z – p – qi)(z – p + qi)