Solve the equation 2x^2 - 5x + 3 = 0.

Step-by-step solution

1. 1

   Use the AC method because the leading coefficient is not $1$. Multiply $a$ by $c$:

   $a\cdot c=2\cdot 3=6$

   and look for two numbers with product $6$ and sum $b=-5$. They are $-2$ and $-3$.

2. 2

   Split the middle term with that pair.

   $2x^2-2x-3x+3=0$

   The split is legitimate because $-2x-3x=-5x$, so nothing has changed.

3. 3

   Group in pairs and factor each group.

   $2x(x-1)-3(x-1)=0$

   Both groups now share the binomial $(x-1)$, which confirms the split was the right one.

4. 4

   Factor out the common binomial and solve.

   $(2x-3)(x-1)=0\quad\Longrightarrow\quad x=\frac{3}{2}\ \text{or}\ x=1$

5. 5

   Check both roots. At $x=1$: $2-5+3=0$ $\checkmark$. At $x=\tfrac32$: $2\left(\tfrac94\right)-\tfrac{15}{2}+3=\tfrac92-\tfrac{15}{2}+3=0$ $\checkmark$. Vieta agrees: the roots sum to $\tfrac52=-\tfrac{b}{a}$ and multiply to $\tfrac32=\tfrac{c}{a}$.