Solve the quadratic equation using factorization: $x^2 - 5x + 6 = 0$.

Step 1: Identify coefficients and find factors

We need to factor the quadratic equation $x^2 - 5x + 6 = 0$. This is in the standard form $ax^2 + bx + c = 0$, where $a=1$, $b=-5$, and $c=6$. We need to find two numbers that multiply to $c$ (which is 6) and add up to $b$ (which is -5).

Step 2: Find the two numbers

The two numbers that satisfy these conditions are -2 and -3, because their product is $(-2) \times (-3) = 6$ and their sum is $(-2) + (-3) = -5$.

Step 3: Rewrite the middle term

Now, we rewrite the middle term, $-5x$, using these two numbers: $-2x - 3x$. This allows us to group terms for factorization.

Step 4: Factor by grouping

We group the terms and factor out the common factors from each pair. From the first two terms, $x^2 - 2x$, we factor out $x$. From the last two terms, $-3x + 6$, we factor out $-3$.

Step 5: Factor out the common binomial

Notice that $(x - 2)$ is a common binomial factor in both terms. We factor it out to get the completely factored form of the quadratic equation.

Step 6: Solve for x

For the product of two factors to be zero, at least one of the factors must be zero. So, we set each factor equal to zero and solve for $x$.

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