I.` p^(2) -15p+56=0`
II.` q =root(3)(512)`

Options

A. If x = y or no relation can be established
B. If`x gt y`
C. If ` x lt y`
D. If` x le y`

Explanation

To solve the given equations and find the relationship between \( P \) and \( Q \), we will follow these steps:

### Step 1: Solve the equation for \( P \)

The equation given is:

\[ P^2 - 15P + 56 = 0 \]

To factor this quadratic equation, we need to find two numbers that multiply to \( 56 \) (the constant term) and add up to \( -15 \) (the coefficient of \( P \)).

### Step 2: Factor the quadratic equation

The factors of \( 56 \) that add up to \( -15 \) are \( -7 \) and \( -8 \). Therefore, we can factor the equation as follows:

\[ (P - 7)(P - 8) = 0 \]

### Step 3: Find the values of \( P \)

Setting each factor to zero gives us:

\[ P - 7 = 0 \quad \Rightarrow \quad P = 7 \]
\[ P - 8 = 0 \quad \Rightarrow \quad P = 8 \]

Thus, the possible values for \( P \) are \( 7 \) and \( 8 \).

### Step 4: Solve the equation for \( Q \)

The equation for \( Q \) is given as:

\[ Q = \sqrt[3]{512} \]

To find \( Q \), we need to calculate the cube root of \( 512 \).

### Step 5: Calculate the cube root of \( 512 \)

We can express \( 512 \) as:

\[ 512 = 8^3 \]

Thus, taking the cube root gives us:

\[ Q = \sqrt[3]{512} = 8 \]

### Step 6: Establish the relationship between \( P \) and \( Q \)

Now we have the values:

- \( P = 7 \) or \( P = 8 \)
- \( Q = 8 \)

We can now determine the relationship between \( P \) and \( Q \):

Since \( P \) can take the values \( 7 \) or \( 8 \), we can say:

\[ P \leq Q \]

### Final Answer

The relationship between \( P \) and \( Q \) is:

\[ P \leq Q \]

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