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If [tex]$p + q = 10$[/tex] and [tex]$p \cdot q = 5$[/tex], then the numerical value of [tex]\frac{p}{q} + \frac{q}{p}[/tex] will be:

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GinnyAnswer
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].
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