There are two quadratic equations. Solve these equations and find the relation between p and q.
`p^2-10p+21=0`
`q^2-10q+24=0`

Options

A. `pltq`
B. `pgtq`
C. `pleq`
D. can't be determined

Answer

To solve the given quadratic equations and find the relation between \( p \) and \( q \), we will follow these steps:

### Step 1: Solve the first quadratic equation \( p^2 - 10p + 21 = 0 \)

We can factor this equation. We need to find two numbers that multiply to \( 21 \) and add up to \( -10 \). The numbers \( -3 \) and \( -7 \) satisfy this condition.

So, we can rewrite the equation as:
\[ p^2 - 3p - 7p + 21 = 0 \]
Now, we can factor by grouping:
\[ p(p - 7) - 3(p - 7) = 0 \]
This gives us:
\[ (p - 3)(p - 7) = 0 \]
Setting each factor to zero, we find:
\[ p - 3 = 0 \quad \Rightarrow \quad p = 3 \]
\[ p - 7 = 0 \quad \Rightarrow \quad p = 7 \]

### Step 2: Solve the second quadratic equation \( q^2 - 10q + 24 = 0 \)

Similarly, we need to find two numbers that multiply to \( 24 \) and add up to \( -10 \). The numbers \( -4 \) and \( -6 \) satisfy this condition.

So, we can rewrite the equation as:
\[ q^2 - 4q - 6q + 24 = 0 \]
Now, we can factor by grouping:
\[ q(q - 4) - 6(q - 4) = 0 \]
This gives us:
\[ (q - 4)(q - 6) = 0 \]
Setting each factor to zero, we find:
\[ q - 4 = 0 \quad \Rightarrow \quad q = 4 \]
\[ q - 6 = 0 \quad \Rightarrow \quad q = 6 \]

### Step 3: Determine the relation between \( p \) and \( q \)

Now we have the values:
- \( p = 3 \) or \( p = 7 \)
- \( q = 4 \) or \( q = 6 \)

We can compare these values:
- For \( p = 3 \): \( q = 4 \) (so \( q > p \))
- For \( p = 7 \): \( q = 6 \) (so \( p > q \))

Since we have one case where \( q > p \) and another case where \( p > q \), we cannot determine a consistent relation between \( p \) and \( q \).

### Final Conclusion

The relation between \( p \) and \( q \) cannot be determined.

---