Question

1. which are the roots of $3x^2 - 5x + 2 = 0$?

$\bigcirc$ $x = 1$, $x = -\frac{2}{3}$
$\bigcirc$ $x = -1, x = -\frac{2}{3}$
$\bigcirc$ $x = 1$, $x = \frac{2}{3}$
$\bigcirc$ $x = -1$, $x = \frac{2}{3}$

Explanation:

Step1: Factor the quadratic equation

We factor $3x^2 -5x +2$ by finding two numbers that multiply to $3\times2=6$ and add to $-5$. These numbers are $-2$ and $-3$. Split the middle term:
$$3x^2 -3x -2x +2 = 0$$
Group and factor:
$$3x(x-1) -2(x-1) = 0$$
$$(3x-2)(x-1) = 0$$

Step2: Solve for $x$

Set each factor equal to 0:
For $x-1=0$: $x=1$
For $3x-2=0$: $3x=2 \implies x=\frac{2}{3}$

Answer:

$x = 1, x = \frac{2}{3}$ (third option)