To solve or expand the given mathematical expression [tex]\(45x^2 + 17\sqrt{15x} + 24\)[/tex], we will go through the steps methodically.
### Step-by-Step Solution
1. Identify the Terms:
The expression is a sum of three terms. Let's break it down:
- The quadratic term: [tex]\(45x^2\)[/tex]
- The mixed term involving a radical: [tex]\(17\sqrt{15x}\)[/tex]
- The constant term: [tex]\(24\)[/tex]
2. Expand the Radical Term:
The term [tex]\(17\sqrt{15x}\)[/tex] can be written considering that [tex]\( \sqrt{15x} \)[/tex] is the square root of the product of 15 and [tex]\(x\)[/tex].
Hence, [tex]\( \sqrt{15x} = \sqrt{15} \cdot \sqrt{x} \)[/tex].
3. Rewriting the Mixed Term:
Substitute [tex]\(\sqrt{15x}\)[/tex] with [tex]\(\sqrt{15} \cdot \sqrt{x}\)[/tex]:
[tex]\[
17\sqrt{15x} = 17 \cdot \sqrt{15} \cdot \sqrt{x}
\][/tex]
4. Combine All the Terms:
Now combine all the terms back into the expression:
[tex]\[
45x^2 + 17 \sqrt{15} \sqrt{x} + 24
\][/tex]
Thus, the final detailed expression, combining all the steps, is:
[tex]\[
45x^2 + 17 \sqrt{15} \sqrt{x} + 24
\][/tex]
This is the expanded and detailed form of the given expression.
