Solve the equation x^2 - 4x + 3 = 0.

Step-by-step solution

1. 1

   Try factoring before the quadratic formula. For a monic quadratic, look for two numbers whose product is the constant $3$ and whose sum is the middle coefficient $-4$. The signs decide the search: a positive product with a negative sum means both numbers are negative.

2. 2

   Find the pair. The only negative factor pair of $3$ is $(-1,-3)$, and it works:

   $(-1)(-3)=3\ \checkmark\qquad -1+(-3)=-4\ \checkmark$

   so

   $x^2-4x+3=(x-1)(x-3)$

3. 3

   Apply the zero-product property. A product of two real numbers is zero only if at least one factor is zero:

   $(x-1)(x-3)=0\qquad\Longrightarrow\qquad x-1=0\ \text{ or }\ x-3=0$

   This step is only valid because the right-hand side is $0$ — it is why moving everything to one side comes first.

4. 4

   Solve each factor.

   $x=1\qquad\text{or}\qquad x=3$

5. 5

   Verify both roots. At $x=1$: $1-4+3=0$ ✓. At $x=3$: $9-12+3=0$ ✓. Vieta also checks out: the roots sum to $4=-b/a$ and multiply to $3=c/a$ ✓.

6. 6

   Note the graph. The parabola opens upward and crosses the axis at $1$ and $3$, so its vertex is at the midpoint $x=2$, where $y=4-8+3=-1$ — the minimum value. That also tells you $x^2-4x+3<0$ exactly on $(1,3)$.