In each of the following questions, two equations (I) and (II) are given. Solve the equations and mark the correct option
I. ` 3 x^(2) + 23 x + 30 = 0 `
II. ` y^(2) + 15 y + 56 = 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 and find the relationship between the values of \(x\) and \(y\), we will follow these steps:

### Step 1: Solve the first equation \(3x^2 + 23x + 30 = 0\)

1. **Factor the quadratic equation**:
We want to express \(3x^2 + 23x + 30\) in a factored form. We can look for two numbers that multiply to \(3 \times 30 = 90\) and add up to \(23\). The numbers \(18\) and \(5\) fit this requirement.

Rewrite the equation:
\[ 3x^2 + 18x + 5x + 30 = 0 \]
Group the terms:
\[ (3x^2 + 18x) + (5x + 30) = 0 \]
Factor by grouping:
\[ 3x(x + 6) + 5(x + 6) = 0 \]
Factor out the common term \((x + 6)\):
\[ (3x + 5)(x + 6) = 0 \]

2. **Find the values of \(x\)**:
Set each factor to zero:
\[ 3x + 5 = 0 \quad \Rightarrow \quad x = -\frac{5}{3} \]
\[ x + 6 = 0 \quad \Rightarrow \quad x = -6 \]

### Step 2: Solve the second equation \(y^2 + 15y + 56 = 0\)

1. **Factor the quadratic equation**:
We need two numbers that multiply to \(56\) and add up to \(15\). The numbers \(7\) and \(8\) work here.

Rewrite the equation:
\[ y^2 + 7y + 8y + 56 = 0 \]
Group the terms:
\[ (y^2 + 7y) + (8y + 56) = 0 \]
Factor by grouping:
\[ y(y + 7) + 8(y + 7) = 0 \]
Factor out the common term \((y + 7)\):
\[ (y + 8)(y + 7) = 0 \]

2. **Find the values of \(y\)**:
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 \(y\)

We have the following values:
- \(x = -6\) and \(x = -\frac{5}{3}\)
- \(y = -8\) and \(y = -7\)

Now we compare:
1. For \(x = -6\):
- Compare \(-6\) with \(-8\) and \(-7\):
- \(-6 > -8\)
- \(-6 > -7\)

2. For \(x = -\frac{5}{3} \approx -1.67\):
- Compare \(-\frac{5}{3}\) with \(-8\) and \(-7\):
- \(-\frac{5}{3} > -8\)
- \(-\frac{5}{3} > -7\)

### Conclusion:
In both cases, \(x\) is greater than \(y\). Therefore, we conclude that:
\[ x > y \]

### Final Answer:
The correct option is \(x > y\).

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