What is the general solution of the equation $2\sin^2 x - 7\sin x + 3 = 0$? - HSC - SSCE Mathematics Extension 1 - Question 6 - 2016 - Paper 1

Applying the quadratic formula:

$y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

where $a = 2$, $b = -7$, and $c = 3$, we calculate:

$b^2 - 4ac = (-7)^2 - 4 \cdot 2 \cdot 3 = 49 - 24 = 25$

Thus, we have:

$y = \frac{7 \pm 5}{4}$

This gives us the two possible solutions:

1. $y = \frac{12}{4} = 3$ (not valid since orall y \in [-1, 1])
2. $y = \frac{2}{4} = \frac{1}{2}$

For $\sin x = \frac{1}{2}$, the general solution is given by:

$x = n\pi + (-1)^n \frac{\pi}{6}$

where $n \in \mathbb{Z}$.

Thus, the general solution is:

$x = n\pi + (-1)^n \frac{\pi}{6}$

This corresponds to option (D): $n\pi + (-1)^n \frac{\pi}{6}$.