Solve the equation x^2 - 5x + 6 = 0.

Step-by-step solution

1. 1

   Deduce the signs of the factors from the coefficients. The constant $+6$ is positive, so both factors share a sign; the middle coefficient $-5$ is negative, so that shared sign is negative. We are looking for two negative numbers.

2. 2

   Find the pair. Product $6$, sum $-5$:

   $(-2)\cdot(-3)=6,\qquad -2+(-3)=-5$

3. 3

   Factor and split.

   $x^2-5x+6=(x-2)(x-3)=0$

   $x-2=0\ \Rightarrow\ x=2,\qquad x-3=0\ \Rightarrow\ x=3$

4. 4

   Check both, and note the graph. At $x=2$: $4-10+6=0$ $\checkmark$; at $x=3$: $9-15+6=0$ $\checkmark$. The parabola $y=x^2-5x+6$ opens upward and crosses the axis at $2$ and $3$, so its vertex sits midway at $x=2.5$ where $y=-0.25$.