<s>[INST] Finding exact solutions of an equation I have the question "Find the exact solutions of the following equation, giving your answer in terms of i and simplifying any surds (where possible): 
2X^2 + 8X + 9 = 0"
Here is my attempt is this correct ? 

 [/INST] Looks correct to me. Here's how I would do it though:
$$
2x^2 + 8x + 9 = 0\implies\\
x^2 + 4x + \frac{9}{2} = 0\implies\\
x^2+2\cdot x\cdot 2+2^2-2^2 +\frac{9}{2} = 0\implies\\
(x+2)^2=\frac{8}{2}-\frac{9}{2}\implies\\
(x+2)^2=-\frac{1}{2}\implies\\
x_{1,2}+2=\pm\sqrt{-\frac{1}{2}}\implies\\
x_{1,2}=-2\pm\sqrt{\frac{1}{2}\cdot (-1)}\implies\\
x_{1,2}=-2\pm\sqrt{\frac{1}{2}\cdot i^2}\implies\\
x_{1,2}=-2\pm\sqrt{\frac{1}{2}}\cdot\sqrt{i^2}\implies\\
x_{1,2}=-2\pm\frac{1}{\sqrt{2}}i\implies\\
x_{1,2}=-2\pm\frac{\sqrt{2}}{2}i
$$
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