Q. \(x^2 – x – 6 = 0\)

We want to solve the equation

\[
x^2 – x – 6 = 0
\]

This is a quadratic equation, so we look for ways to factor it (or use the quadratic formula). Since it has nice integers, factoring is the fastest method.

Step 1: Identify the coefficients.

Compare

\[
x^2 – x – 6 = 0
\]

with the general form

\[
ax^2 + bx + c = 0
\]

Here:

• \(a = 1\)
• \(b = -1\)
• \(c = -6\)

Step 2: Factor the quadratic.

Factor the expression \(x^2 – x – 6\) into two binomials of the form

\[
(x + m)(x + n)
\]

where \(m\) and \(n\) are integers.

For the product to match:

• \(m \cdot n = -6\)
• \(m + n = -1\) (because the middle term is \(-x\))

Step 3: Find integers that satisfy both conditions.

List factor pairs of \(-6\):

• \(-1\) and \(6\) gives sum \(5\)
• \(1\) and \(-6\) gives sum \(-5\)
• \(-2\) and \(3\) gives sum \(1\)
• \(2\) and \(-3\) gives sum \(-1\)

The pair \(2\) and \(-3\) works because:

• \(2 \cdot (-3) = -6\)
• \(2 + (-3) = -1\)

Step 4: Write the factored form.

\[
x^2 – x – 6 = (x + 2)(x – 3)
\]

Step 5: Set each factor equal to zero.

Since the product is zero:

\[
(x + 2)(x – 3) = 0
\]

we get two equations:

\[
x + 2 = 0
\]
\[
x – 3 = 0
\]

Step 6: Solve each equation.

From \(x + 2 = 0\):

\[
x = -2
\]

From \(x – 3 = 0\):

\[
x = 3
\]

Final Answer:

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