There are two quadratic equations. Solve these equations and find the relation between p and q.
`p^2-7p+10=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 \( p^2 - 7p + 10 = 0 \)

We will factor the quadratic equation. We need to find two numbers that multiply to \( 10 \) (the constant term) and add up to \( -7 \) (the coefficient of \( p \)).

1. The numbers that satisfy this are \( -5 \) and \( -2 \).
2. We can rewrite the equation as:
\[ p^2 - 5p - 2p + 10 = 0 \]
3. Now, we group the terms:
\[ (p^2 - 5p) + (-2p + 10) = 0 \]
4. Factor out the common terms:
\[ p(p - 5) - 2(p - 5) = 0 \]
5. This can be factored as:
\[ (p - 5)(p - 2) = 0 \]
6. Setting each factor to zero gives us:
\[ p - 5 = 0 \quad \Rightarrow \quad p = 5 \]
\[ p - 2 = 0 \quad \Rightarrow \quad p = 2 \]

So, the solutions for \( p \) are \( p = 2 \) and \( p = 5 \).

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

Similarly, we will factor this quadratic equation. We need to find two numbers that multiply to \( 24 \) and add up to \( -10 \).

1. The numbers that satisfy this are \( -6 \) and \( -4 \).
2. We can rewrite the equation as:
\[ q^2 - 6q - 4q + 24 = 0 \]
3. Now, we group the terms:
\[ (q^2 - 6q) + (-4q + 24) = 0 \]
4. Factor out the common terms:
\[ q(q - 6) - 4(q - 6) = 0 \]
5. This can be factored as:
\[ (q - 6)(q - 4) = 0 \]
6. Setting each factor to zero gives us:
\[ q - 6 = 0 \quad \Rightarrow \quad q = 6 \]
\[ q - 4 = 0 \quad \Rightarrow \quad q = 4 \]

So, the solutions for \( q \) are \( q = 4 \) and \( q = 6 \).

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

Now we have the values:
- \( p = 2 \) and \( p = 5 \)
- \( q = 4 \) and \( q = 6 \)

We will compare the values of \( p \) and \( q \):
- For \( q = 4 \):
- \( 4 > 2 \) (so \( q \) is greater than the smaller \( p \))
- \( 4 < 5 \) (so \( q \) is less than the larger \( p \))

- For \( q = 6 \):
- \( 6 > 2 \) (so \( q \) is greater than the smaller \( p \))
- \( 6 > 5 \) (so \( q \) is greater than the larger \( p \))

### Conclusion

From the above comparisons, we can conclude that:
- \( q = 4 \) is between the two values of \( p \).
- \( q = 6 \) is greater than both values of \( p \).

Thus, we cannot definitively determine a specific relation between \( p \) and \( q \) since \( q \) can be either between or greater than the values of \( p \).