Factor x^2 - 3x + 2, then use the factorisation to solve x^2 - 3x + 2 = 0.

Step-by-step solution

1. 1

   Read the signs off the coefficients first. The constant $+2$ is positive, so the two factors share a sign; the middle coefficient $-3$ is negative, so that shared sign is negative. We are looking for two negative numbers.

2. 2

   Find the pair. With a leading coefficient of $1$, the numbers must multiply to $2$ and add to $-3$:

   $(-1)\cdot(-2)=2,\qquad -1+(-2)=-3$

   so the pair is $-1$ and $-2$.

3. 3

   Write the factorisation.

   $x^2-3x+2=(x-1)(x-2)$

4. 4

   Solve the equation with the zero-product property. A product is zero exactly when one factor is zero:

   $x-1=0\ \Rightarrow\ x=1,\qquad x-2=0\ \Rightarrow\ x=2$

   Note the sign flip: the factor $(x-1)$ gives the positive root $1$.

5. 5

   Check both the factorisation and the roots. Expanding: $(x-1)(x-2)=x^2-2x-x+2=x^2-3x+2$ $\checkmark$. Substituting: at $x=1$, $1-3+2=0$ $\checkmark$; at $x=2$, $4-6+2=0$ $\checkmark$. Vieta agrees too, since the roots sum to $3=-\tfrac{b}{a}$ and multiply to $2=\tfrac{c}{a}$.