In each question two equations numbered (I) and (II) are given . You have to solve both the equations and mark appropriate answer
I. ` x^(2) + 4 x - 45 = 0 `
II. ` y^(2) - 13 y + 40 = 0 `

Options

A. If x= y or no relation
B. If ` x gt 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 proceed to equation II.

### Step 1: Solve Equation I
The first equation is:
\[ x^2 + 4x - 45 = 0 \]

To factor this quadratic equation, we need to find two numbers that multiply to \(-45\) (the constant term) and add up to \(4\) (the coefficient of \(x\)).

The numbers that satisfy these conditions are \(9\) and \(-5\) because:
\[ 9 \times (-5) = -45 \]
\[ 9 + (-5) = 4 \]

Now, we can rewrite the equation as:
\[ x^2 + 9x - 5x - 45 = 0 \]

Next, we group the terms:
\[ (x^2 + 9x) + (-5x - 45) = 0 \]

Factoring by grouping:
\[ x(x + 9) - 5(x + 9) = 0 \]

Now, factor out \((x + 9)\):
\[ (x + 9)(x - 5) = 0 \]

Setting each factor to zero gives us:
1. \( x + 9 = 0 \) → \( x = -9 \)
2. \( x - 5 = 0 \) → \( x = 5 \)

### Step 2: Solve Equation II
The second equation is:
\[ y^2 - 13y + 40 = 0 \]

We need to find two numbers that multiply to \(40\) and add up to \(-13\). The numbers that satisfy these conditions are \(-8\) and \(-5\) because:
\[ -8 \times -5 = 40 \]
\[ -8 + (-5) = -13 \]

Now, we can rewrite the equation as:
\[ y^2 - 8y - 5y + 40 = 0 \]

Next, we group the terms:
\[ (y^2 - 8y) + (-5y + 40) = 0 \]

Factoring by grouping:
\[ y(y - 8) - 5(y - 8) = 0 \]

Now, factor out \((y - 8)\):
\[ (y - 8)(y - 5) = 0 \]

Setting each factor to zero gives us:
1. \( y - 8 = 0 \) → \( y = 8 \)
2. \( y - 5 = 0 \) → \( y = 5 \)

### Step 3: Compare Values of x and y
Now we have the solutions:
- From Equation I: \( x = -9 \) or \( x = 5 \)
- From Equation II: \( y = 8 \) or \( y = 5 \)

We can compare the values:
1. For \( x = -9 \):
- \( -9 < 5 \)
- \( -9 < 8 \)

2. For \( x = 5 \):
- \( 5 = 5 \)
- \( 5 < 8 \)

### Conclusion
From the comparisons, we can conclude:
- \( x \) can be less than \( y \) or equal to \( y \).
- Therefore, the relation can be expressed as \( x \leq y \).

### Final Answer
The relation between \( x \) and \( y \) is:
\[ x \leq y \]