Two equations (I) and (II) are given in each question. On the basis of these equations you have to decide the relation between x and y and give answer
I. ` 2 x^(2) + 23 x +63 = 0 `
II. ` 4y ^(2) + 19 y + 21 = 0 `

Options

A. A)If ` x gt y `
B. B)If ` x lt y `
C. C) If ` x ge y`
D. D)If ` x le 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 \( 2x^2 + 23x + 63 = 0 \)

To solve this quadratic equation, we can factor it. We need to find two numbers that multiply to \( 2 \times 63 = 126 \) and add to \( 23 \).

The numbers that satisfy these conditions are \( 14 \) and \( 9 \). So we can rewrite the equation as:

\[ 2x^2 + 14x + 9x + 63 = 0 \]

Now, we can group the terms:

\[ (2x^2 + 14x) + (9x + 63) = 0 \]

Factoring out the common terms:

\[ 2x(x + 7) + 9(x + 7) = 0 \]

Now, we can factor out \( (x + 7) \):

\[ (x + 7)(2x + 9) = 0 \]

Setting each factor to zero gives us:

1. \( x + 7 = 0 \) → \( x = -7 \)
2. \( 2x + 9 = 0 \) → \( x = -\frac{9}{2} \)

### Step 2: Solve the second equation \( 4y^2 + 19y + 21 = 0 \)

Similarly, we will factor this quadratic equation. We need to find two numbers that multiply to \( 4 \times 21 = 84 \) and add to \( 19 \).

The numbers that satisfy these conditions are \( 12 \) and \( 7 \). So we can rewrite the equation as:

\[ 4y^2 + 12y + 7y + 21 = 0 \]

Now, we can group the terms:

\[ (4y^2 + 12y) + (7y + 21) = 0 \]

Factoring out the common terms:

\[ 4y(y + 3) + 7(y + 3) = 0 \]

Now, we can factor out \( (y + 3) \):

\[ (y + 3)(4y + 7) = 0 \]

Setting each factor to zero gives us:

1. \( y + 3 = 0 \) → \( y = -3 \)
2. \( 4y + 7 = 0 \) → \( y = -\frac{7}{4} \)

### Step 3: Compare the values of \( x \) and \( y \)

Now we have the values:

- For \( x \): \( -7 \) and \( -\frac{9}{2} \)
- For \( y \): \( -3 \) and \( -\frac{7}{4} \)

We will compare these values:

1. Compare \( x = -7 \) with \( y = -3 \):
\[ -7 < -3 \]

2. Compare \( x = -7 \) with \( y = -\frac{7}{4} \):
\[ -7 < -\frac{7}{4} \]

3. Compare \( x = -\frac{9}{2} \) with \( y = -3 \):
\[ -\frac{9}{2} = -4.5 < -3 \]

4. Compare \( x = -\frac{9}{2} \) with \( y = -\frac{7}{4} \):
\[ -\frac{9}{2} = -4.5 < -\frac{7}{4} = -1.75 \]

### Conclusion

From all comparisons, we find that:

\[ x < y \]

Thus, the final relationship is:

\[ \text{Answer: } x < y \]