Q. \(\,x^2-8x+15\,\)

We want to simplify the expression \(x^2 – 8x + 15\). A common goal is to factor a quadratic when possible.

Step 1: Identify the quadratic form.
We have a quadratic in standard form:

\[
x^2 – 8x + 15
\]

Step 2: Factor using the form \((x-a)(x-b)\).
If we can factor it, it will look like:

\[
(x-a)(x-b)
\]

Expanding gives:

\[
(x-a)(x-b) = x^2 – (a+b)x + ab
\]

Step 3: Match coefficients with the given quadratic.
Comparing:

• The coefficient of \(x^2\) is already \(1\), which matches.
• We need \(a+b = 8\) because the middle term is \(-8x\).
• We need \(ab = 15\) because the constant term is \(15\).

Step 4: Find numbers \(a\) and \(b\) that satisfy both conditions.
We need two numbers whose product is \(15\) and whose sum is \(8\).

• \(3 \cdot 5 = 15\)
• \(3 + 5 = 8\)

So we can take \(a = 3\) and \(b = 5\).

Step 5: Write the factored form.
Since the quadratic is \(x^2 – 8x + 15\), the factorization is:

\[
(x-3)(x-5)
\]

Step 6: (Optional check) Expand to confirm.
Expand:

\[
(x-3)(x-5) = x^2 – 5x – 3x + 15 = x^2 – 8x + 15
\]

This matches the original expression.

Final Answer:

\[
x^2 – 8x + 15 = (x-3)(x-5)
\]