Solve the equation x² − 5x + 6 = 0

Answer: \(x=2\) or \(x=3\).

How to solve

We need two numbers whose product is \(6\) and whose sum is \(5\), because

\[ x^2-5x+6=(x-2)(x-3). \]

Indeed,

\[ -2\cdot(-3)=6,\qquad -2+(-3)=-5. \]

So the equation becomes

\[ (x-2)(x-3)=0. \]

A product is zero only when at least one factor is zero:

\[ x-2=0 \quad\text{or}\quad x-3=0, \]

therefore

\[ x=2 \quad\text{or}\quad x=3. \]

Check

For \(x=2\):

\[ 2^2-5\cdot2+6=4-10+6=0. \]

For \(x=3\):

\[ 3^2-5\cdot3+6=9-15+6=0. \]

Both values work. The key idea is to factor the quadratic into two brackets, then set each bracket equal to zero.