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]