One root of the equation \({x^2} + ax + b = 0\) is \(2 + 3{\text{i}}\) where \(a,{\text{ }}b \in \mathbb{R}\). Find the value of \(a\) and the value of \(b\).

METHOD 1

substituting

\( - 5 + 12{\text{i}} + a(2 + 3{\text{i}}) + b = 0\)     (A1)

equating real or imaginary parts     (M1)

\(12 + 3a = 0 \Rightarrow a =  - 4\)     A1

\( - 5 + 2a + b = 0 \Rightarrow b = 13\)     A1

METHOD 2

other root is \(2 - 3{\text{i}}\)     (A1)

considering either the sum or product of roots or multiplying factors     (M1)

\(4 =  - a\) (sum of roots) so \(a =  - 4\)     A1

\(13 = b\) (product of roots)     A1

[4 marks]