In each of these questions, two equations I. and II. are given. You have to solve both the equations and give answer
I. ` 3 x^(2) - 60 x + 2 88 = 0 `
II. ` 4 y ^(2) - 50 y + 156 = 0 `

Options

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

Answer

To solve the given equations step by step, let's start with the first equation.

### Step 1: Solve the first equation
The first equation is:
\[ 3x^2 - 60x + 288 = 0 \]

We can simplify this equation by dividing all terms by 3:
\[ x^2 - 20x + 96 = 0 \]

Next, we will factor this quadratic equation. We need two numbers that multiply to 96 and add up to -20. The numbers -12 and -8 satisfy these conditions:
\[ (x - 12)(x - 8) = 0 \]

Setting each factor to zero gives us the solutions for \( x \):
1. \( x - 12 = 0 \) → \( x = 12 \)
2. \( x - 8 = 0 \) → \( x = 8 \)

### Step 2: Solve the second equation
The second equation is:
\[ 4y^2 - 50y + 156 = 0 \]

We can simplify this equation by dividing all terms by 2:
\[ 2y^2 - 25y + 78 = 0 \]

Now, we will factor this quadratic equation. We need two numbers that multiply to \( 2 \times 78 = 156 \) and add up to -25. The numbers -13 and -12 satisfy these conditions:
\[ (2y - 13)(y - 6) = 0 \]

Setting each factor to zero gives us the solutions for \( y \):
1. \( 2y - 13 = 0 \) → \( y = \frac{13}{2} \)
2. \( y - 6 = 0 \) → \( y = 6 \)

### Step 3: Summarize the solutions
Now we have the solutions:
- For \( x \): \( 12, 8 \)
- For \( y \): \( 6, \frac{13}{2} \)

### Step 4: Compare the values of \( x \) and \( y \)
Now we will compare the values of \( x \) and \( y \):
1. For \( x = 12 \) and \( y = 6 \): \( 12 > 6 \)
2. For \( x = 8 \) and \( y = 6 \): \( 8 > 6 \)
3. For \( x = 12 \) and \( y = \frac{13}{2} \) (which is 6.5): \( 12 > \frac{13}{2} \)
4. For \( x = 8 \) and \( y = \frac{13}{2} \): \( 8 > \frac{13}{2} \) (false, since \( 8 < 6.5 \))

### Conclusion
From the comparisons, we can conclude that:
- In all cases where \( y = 6 \), \( x \) is greater than \( y \).
- In the case where \( y = \frac{13}{2} \), \( x = 12 \) is greater, but \( x = 8 \) is not.

Thus, the overall relation is:
\[ x > y \]

### Final Answer
The correct answer is that \( x \) is greater than \( y \).

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