Q. \(x^2 – 6x + 5\)

We want to work with the expression \(x^2 – 6x + 5\). A common goal of a quadratic like this is to factor it, if possible, because factoring helps us find zeros and simplify the expression.

Step 1: Understand the quadratic form.

The expression is a quadratic in \(x\):

\[
x^2 – 6x + 5
\]

Here, we have \(a = 1\), \(b = -6\), and \(c = 5\).

Step 2: Find two numbers that multiply to \(c\) and add to \(b\).

We look for numbers \(m\) and \(n\) such that:

\[
m \cdot n = 5
\]

and

\[
m + n = -6
\]

Now list the factor pairs of \(5\): \(1 \cdot 5\) and \((-1) \cdot (-5)\). Their sums are \(1 + 5 = 6\) and \((-1) + (-5) = -6\).

So the correct pair is \(-1\) and \(-5\).

Step 3: Factor the quadratic using those two numbers.

Rewrite the middle term \(-6x\) as the sum of two terms that match \(-1\) and \(-5\):

\[
x^2 – 6x + 5 = x^2 – 5x – x + 5
\]

The expression is now ready to factor by grouping.

Step 4: Factor by grouping.

Group the first two terms and the last two terms:

\[
(x^2 – 5x) + (-x + 5)
\]

Factor out the greatest common factor from each group.

First group:

\[
x^2 – 5x = x(x – 5)
\]

Second group:

\[
-x + 5 = -1(x – 5)
\]

So the whole expression becomes:

\[
x(x – 5) – 1(x – 5)
\]

Now factor out the common factor \((x – 5)\):

\[
(x – 5)(x – 1)
\]

Thus, the quadratic factors as:

\[
x^2 – 6x + 5 = (x – 1)(x – 5)
\]

Final Answer:

\[
x^2 – 6x + 5 = (x – 1)(x – 5)
\]