X²−3x+2=0,use factorisation method

The given quadratic equation is in the form $ax^2 + bx + c = 0$, where $a=1$, $b=-3$, and $c=2$.

We need to find two numbers that multiply to $c$ (which is 2) and add up to $b$ (which is -3).
Let these numbers be $p$ and $q$.
$p \times q = 2$
$p + q = -3$

The numbers are -1 and -2, because $(-1) \times (-2) = 2$ and $(-1) + (-2) = -3$.

Substitute $-3x$ with $-1x - 2x$:
$x^2 - x - 2x + 2 = 0$

Group the terms and factor out common factors:
$(x^2 - x) + (-2x + 2) = 0$
$x(x - 1) - 2(x - 1) = 0$

Notice that $(x - 1)$ is a common factor:
$(x - 1)(x - 2) = 0$

Set each factor equal to zero and solve for $x$:
$x - 1 = 0 \implies x = 1$
$x - 2 = 0 \implies x = 2$

The solutions are:
$x_1 = \boxed{1}$
$x_2 = \boxed{2}$