Answer
$x=\left\{ 2,3 \right\}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given eqution, $
x^2-5x+6=0
,$ express the equation in factored form. Then equate each factor to zero (Zero Product Property). Finally, solve each equation.
$\bf{\text{Solution Details:}}$
In the trinomial expression above, the value of $c$ is $
6
$ and the value of $b$ is $
-5
.$
The possible pairs of integers whose product is $c$ are
\begin{array}{l}\require{cancel}
\{ 1,6 \}, \{ 2,3 \},
\\
\{ -1,-6 \}, \{ -2,-3 \}
.\end{array}
Among these pairs, the one that gives a sum of $b$ is $\{
-2,-3
\}.$ Hence, the factored form of the equation above is
\begin{array}{l}\require{cancel}
(x-2)(x-3)=0
.\end{array}
Equating each factor to zero (Zero Product Property), the solutions to the equation above are
\begin{array}{l}\require{cancel}
x-2=0
\\\\\text{OR}\\\\
x-3=0
.\end{array}
Solving each equation results to
\begin{array}{l}\require{cancel}
x-2=0
\\\\
x=2
\\\\\text{OR}\\\\
x-3=0
\\\\
x=3
.\end{array}
Hence, $
x=\left\{ 2,3 \right\}
.$
