In each of the following questions, two equations (I) and (II) are given. Solve the equations and mark the correct option
I. ` x^(2) + 17 x + 72 = 0`
II. ` y^(2) + 13 y + 42 = 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 equations given in the question, we will follow these steps:

### Step 1: Solve the first equation \( x^2 + 17x + 72 = 0 \)

We will factor the quadratic equation. We need to find two numbers that multiply to \( 72 \) (the constant term) and add up to \( 17 \) (the coefficient of \( x \)).

The numbers that satisfy these conditions are \( 9 \) and \( 8 \) because:
- \( 9 \times 8 = 72 \)
- \( 9 + 8 = 17 \)

Thus, we can rewrite the equation as:
\[ x^2 + 9x + 8x + 72 = 0 \]

### Step 2: Group the terms

Now, we can group the terms:
\[ (x^2 + 9x) + (8x + 72) = 0 \]

### Step 3: Factor by grouping

Now, we factor out the common terms:
\[ x(x + 9) + 8(x + 9) = 0 \]

This can be factored further as:
\[ (x + 9)(x + 8) = 0 \]

### Step 4: Solve for \( x \)

Setting each factor equal to zero gives us:
1. \( x + 9 = 0 \) → \( x = -9 \)
2. \( x + 8 = 0 \) → \( x = -8 \)

So, the solutions for \( x \) are \( x = -9 \) and \( x = -8 \).

### Step 5: Solve the second equation \( y^2 + 13y + 42 = 0 \)

Similarly, we will factor the second quadratic equation. We need to find two numbers that multiply to \( 42 \) and add up to \( 13 \).

The numbers that satisfy these conditions are \( 7 \) and \( 6 \) because:
- \( 7 \times 6 = 42 \)
- \( 7 + 6 = 13 \)

Thus, we can rewrite the equation as:
\[ y^2 + 7y + 6y + 42 = 0 \]

### Step 6: Group the terms

Now, we can group the terms:
\[ (y^2 + 7y) + (6y + 42) = 0 \]

### Step 7: Factor by grouping

Now, we factor out the common terms:
\[ y(y + 7) + 6(y + 7) = 0 \]

This can be factored further as:
\[ (y + 7)(y + 6) = 0 \]

### Step 8: Solve for \( y \)

Setting each factor equal to zero gives us:
1. \( y + 7 = 0 \) → \( y = -7 \)
2. \( y + 6 = 0 \) → \( y = -6 \)

So, the solutions for \( y \) are \( y = -7 \) and \( y = -6 \).

### Step 9: Compare the values of \( x \) and \( y \)

Now we have the values:
- For \( x \): \( -9, -8 \)
- For \( y \): \( -7, -6 \)

We can compare:
1. If \( x = -9 \), then \( -9 < -7 \) (which means \( x < y \))
2. If \( x = -8 \), then \( -8 < -6 \) (which also means \( x < y \))

### Final Conclusion

From both comparisons, we conclude that \( x < y \).