Q. \(x^2 – 8x + 12\)

We want to analyze the expression \(x^2 – 8x + 12\). A common goal with a quadratic like this is to factor it (or, equivalently, find its roots). I will factor it step by step.

Step 1: Identify the quadratic form

We have the quadratic polynomial in the standard form:

\[x^2 – 8x + 12\]

Here:

\(a = 1\), \(b = -8\), and \(c = 12\).

Step 2: Find two numbers that multiply to \(ac\)

We need two integers whose product is:

\[ac = 1 \cdot 12 = 12\]

So we look for two numbers that multiply to \(12\).

Step 3: Find two numbers that add to \(b\)

We also need those same two numbers to add to:

\[b = -8\]

So we want two numbers that:

• Multiply to \(12\)
• Add to \(-8\)

The numbers are \(-6\) and \(-2\), because:

\[(-6)(-2) = 12\]

\[(-6) + (-2) = -8\]

Step 4: Split the middle term and factor by grouping

Rewrite the polynomial by splitting \(-8x\) into two terms using \(-6x\) and \(-2x\):

\[x^2 – 8x + 12 = x^2 – 6x – 2x + 12\]

Now group terms:

\[x^2 – 6x – 2x + 12 = \left(x^2 – 6x\right) + \left(-2x + 12\right)\]

Step 5: Factor each group

Factor the first group \(x^2 – 6x\):

\[x^2 – 6x = x(x – 6)\]

Factor the second group \(-2x + 12\):

\[-2x + 12 = -2(x – 6)\]

Step 6: Factor out the common binomial

Now the expression becomes:

\[x(x – 6) – 2(x – 6)\]

Factor out \((x – 6)\):

\[(x – 6)\left(x – 2\right)\]

Final Answer

\[x^2 – 8x + 12 = (x – 6)(x – 2)\]