I. ` 4x^(2) = 25" " ` II.` 2y^(2) - 13y + 21 = 0`

Options

A. if ` x lt y`
B. if ` x gt y`
C. if ` x = y`
D. if `x ge y`

Explanation

Let's solve the given equations step by step.

### Step 1: Solve for x in the equation \( 4x^2 = 25 \)

To isolate \( x^2 \), we divide both sides by 4:

\[ x^2 = \frac{25}{4} \]

### Step 2: Take the square root of both sides

Taking the square root gives us two possible values for \( x \):

\[ x = \pm \sqrt{\frac{25}{4}} = \pm \frac{5}{2} \]

Thus, the values of \( x \) are:

\[ x = \frac{5}{2} \quad \text{and} \quad x = -\frac{5}{2} \]

### Step 3: Solve for y in the equation \( 2y^2 - 13y + 21 = 0 \)

We can use the factorization method. First, we need to find two numbers that multiply to \( 2 \times 21 = 42 \) and add up to \(-13\).

After testing combinations, we find:

\[ -6 \quad \text{and} \quad -7 \]

### Step 4: Rewrite the equation using the found numbers

We can rewrite the equation as:

\[ 2y^2 - 6y - 7y + 21 = 0 \]

### Step 5: Factor by grouping

Grouping the terms gives us:

\[ (2y^2 - 6y) + (-7y + 21) = 0 \]

Factoring out common terms:

\[ 2y(y - 3) - 7(y - 3) = 0 \]

Now we can factor out \( (y - 3) \):

\[ (2y - 7)(y - 3) = 0 \]

### Step 6: Set each factor to zero

Setting each factor to zero gives us:

1. \( 2y - 7 = 0 \) → \( y = \frac{7}{2} \)
2. \( y - 3 = 0 \) → \( y = 3 \)

Thus, the values of \( y \) are:

\[ y = \frac{7}{2} \quad \text{and} \quad y = 3 \]

### Step 7: Compare the values of x and y

Now we have the values:

- \( x = \frac{5}{2} \) or \( x = -\frac{5}{2} \)
- \( y = \frac{7}{2} \) or \( y = 3 \)

### Step 8: Determine the relationship between x and y

- For \( x = \frac{5}{2} \) (which is 2.5):
- \( y = 3 \) (which is greater than 2.5)
- \( y = \frac{7}{2} \) (which is 3.5, also greater than 2.5)

- For \( x = -\frac{5}{2} \) (which is -2.5):
- Both values of \( y \) (3 and 3.5) are greater than -2.5.

### Conclusion

In both cases, we find that \( y \) is greater than \( x \). Therefore, the relationship is:

\[ y > x \]

---