In each of these questions, two equations I. and II are given. You have to solve both the equations and give answer
I.` x^(3) = 216 `
II. ` 2 y ^(2) - 25 y + 78 = 0 `

Options

A. If ` x gt y `
B. If ` x ge y`
C. If ` x lt y `
D. If ` x le y `

Answer

To solve the given equations step by step, we will start with Equation I and then move on to Equation II.

### Step 1: Solve Equation I
The first equation is:
\[ x^3 = 216 \]

To find \( x \), we take the cube root of both sides:
\[ x = \sqrt[3]{216} \]

Calculating the cube root:
\[ x = 6 \]

### Step 2: Solve Equation II
The second equation is:
\[ 2y^2 - 25y + 78 = 0 \]

To solve this quadratic equation, we can use the quadratic formula:
\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 2 \), \( b = -25 \), and \( c = 78 \).

Calculating the discriminant:
\[ b^2 - 4ac = (-25)^2 - 4 \cdot 2 \cdot 78 \]
\[ = 625 - 624 \]
\[ = 1 \]

Now substituting back into the quadratic formula:
\[ y = \frac{-(-25) \pm \sqrt{1}}{2 \cdot 2} \]
\[ = \frac{25 \pm 1}{4} \]

Calculating the two possible values for \( y \):
1. \( y = \frac{26}{4} = 6.5 \)
2. \( y = \frac{24}{4} = 6 \)

Thus, the solutions for \( y \) are:
\[ y = 6.5 \quad \text{and} \quad y = 6 \]

### Step 3: Compare the values of \( x \) and \( y \)
From the solutions we found:
- \( x = 6 \)
- \( y = 6 \) or \( y = 6.5 \)

Now we compare \( x \) and \( y \):
1. When \( y = 6 \): \( x = y \)
2. When \( y = 6.5 \): \( x < y \)

### Conclusion
The relationship between \( x \) and \( y \) can be summarized as:
\[ x \leq y \]

### Final Answer
The relationship is \( x \leq y \).

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