To find the numerical value of [tex]\(\frac{p}{q} + \frac{q}{p}\)[/tex] given that [tex]\(p + q = 10\)[/tex] and [tex]\(pq = 5\)[/tex], we will follow these steps:
1. Step 1: Identity to Use
We use the identity:
[tex]\[
\frac{p}{q} + \frac{q}{p} = \frac{p^2 + q^2}{pq}
\][/tex]
2. Step 2: Express [tex]\(p^2 + q^2\)[/tex] in terms of [tex]\(p + q\)[/tex] and [tex]\(pq\)[/tex]
We know the identity:
[tex]\[
(p + q)^2 = p^2 + q^2 + 2pq
\][/tex]
We can rearrange this to isolate [tex]\(p^2 + q^2\)[/tex]:
[tex]\[
p^2 + q^2 = (p + q)^2 - 2pq
\][/tex]
3. Step 3: Substitute the Given Values
Substitute [tex]\(p + q = 10\)[/tex] and [tex]\(pq = 5\)[/tex] into the equation:
[tex]\[
p^2 + q^2 = (10)^2 - 2 \cdot 5
\][/tex]
Calculate each term:
[tex]\[
(10)^2 = 100 \quad \text{and} \quad 2 \cdot 5 = 10
\][/tex]
Therefore:
[tex]\[
p^2 + q^2 = 100 - 10 = 90
\][/tex]
4. Step 4: Use the Identity to Find [tex]\(\frac{p}{q} + \frac{q}{p}\)[/tex]
Substitute [tex]\(p^2 + q^2 = 90\)[/tex] and [tex]\(pq = 5\)[/tex] into the identity:
[tex]\[
\frac{p}{q} + \frac{q}{p} = \frac{p^2 + q^2}{pq} = \frac{90}{5}
\][/tex]
Now calculate the fraction:
[tex]\[
\frac{90}{5} = 18
\][/tex]
Thus, the numerical value of [tex]\(\frac{p}{q} + \frac{q}{p}\)[/tex] is [tex]\(18\)[/tex].
