Q. \(x^2 – 5x – 6\)

We want to solve the expression \(x^2 – 5x – 6\). A standard and useful goal with a quadratic like this is to factor it.

Step 1: Identify the quadratic form.

The expression is already in quadratic form:

\[
x^2 – 5x – 6
\]

It matches \(ax^2 + bx + c\) with \(a = 1\), \(b = -5\), and \(c = -6\).

Step 2: Factor \(x^2 – 5x – 6\).

We look for two numbers that:

1. Multiply to \(a \cdot c = 1 \cdot (-6) = -6\).

2. Add to \(b = -5\).

Now list factor pairs of \(-6\): \(-1\) and \(6\), and \(1\) and \(-6\).

Check their sums:

\(-1 + (-6) = -7\) (not \(-5\)).

\(1 + (-6) = -5\) (this matches!).

Step 3: Write the factored form.

Using \(1\) and \(-6\), we split the middle term \(-5x\) into two parts:

\[
x^2 – 5x – 6 = x^2 + x – 6x – 6
\]

Step 4: Factor by grouping.

Group the terms:

\[
x^2 + x – 6x – 6
\]

Group as:

\[
(x^2 + x) + (-6x – 6)
\]

Factor each group:

\[
x(x + 1) – 6(x + 1)
\]

Now factor out the common factor \((x + 1)\):

\[
(x + 1)(x – 6)
\]

Final Answer:

Therefore, the expression factors as:

\[
x^2 – 5x – 6 = (x + 1)(x – 6)
\]