Each of the following questions is provided with 2 statements i.e. Statement I & Statement II. You have the solve them and mark the correct option
I. ` x^(2)- x - 30 = 0 `
II. ` y^(2) - 15 y + 56 = 0 `

Options

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

Answer

To solve the given equations and determine the relationship between \( x \) and \( y \), we will follow these steps:

### Step 1: Solve the first equation \( x^2 - x - 30 = 0 \)
We will factor the quadratic equation.

1. Rewrite the equation:
\[ x^2 - x - 30 = 0 \]

2. Factor the equation:
We need two numbers that multiply to \(-30\) and add to \(-1\). The numbers are \(-6\) and \(5\).
\[ (x - 6)(x + 5) = 0 \]

3. Set each factor to zero:
\[ x - 6 = 0 \quad \Rightarrow \quad x = 6 \]
\[ x + 5 = 0 \quad \Rightarrow \quad x = -5 \]

### Step 2: Solve the second equation \( y^2 - 15y + 56 = 0 \)
We will also factor this quadratic equation.

1. Rewrite the equation:
\[ y^2 - 15y + 56 = 0 \]

2. Factor the equation:
We need two numbers that multiply to \(56\) and add to \(-15\). The numbers are \(-8\) and \(-7\).
\[ (y - 8)(y - 7) = 0 \]

3. Set each factor to zero:
\[ y - 8 = 0 \quad \Rightarrow \quad y = 8 \]
\[ y - 7 = 0 \quad \Rightarrow \quad y = 7 \]

### Step 3: Compare the values of \( x \) and \( y \)
Now we have the values:
- For \( x \): \( 6 \) and \( -5 \)
- For \( y \): \( 8 \) and \( 7 \)

1. Compare \( x = 6 \) with \( y = 8 \) and \( y = 7 \):
- \( 6 < 8 \)
- \( 6 < 7 \)

2. Compare \( x = -5 \) with \( y = 8 \) and \( y = 7 \):
- \( -5 < 8 \)
- \( -5 < 7 \)

### Conclusion
In both cases, we find that \( x < y \). Therefore, the relationship we have established is:
\[ x < y \]

### Final Answer
The correct option is \( \text{B} \) (if \( x < y \) is one of the options).

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