In each question two equations numbered I and II are given . you have to solve both the equations and mark the appropriate option
I `4x^2-17x+18=0`
II `3y^2-8y+4=0`

Options

A. `xgty`
B. The relation between x and y cannot be established.
C. `xlty`
D. `xgey`

Answer

To solve the equations given in the question, we will follow these steps:

### Step 1: Solve the first equation \(4x^2 - 17x + 18 = 0\)

1. **Identify coefficients**: Here, \(a = 4\), \(b = -17\), and \(c = 18\).
2. **Calculate the product \(ac\)**:
\[ ac = 4 \times 18 = 72 \]
3. **Find two numbers that multiply to \(ac\) (72) and add to \(b\) (-17)**: The numbers are \(-9\) and \(-8\).
4. **Rewrite the equation**:
\[ 4x^2 - 9x - 8x + 18 = 0 \]
5. **Factor by grouping**:
\[ (4x^2 - 9x) + (-8x + 18) = 0 \]
\[ x(4x - 9) - 2(4x - 9) = 0 \]
\[ (4x - 9)(x - 2) = 0 \]
6. **Set each factor to zero**:
\[ 4x - 9 = 0 \quad \Rightarrow \quad x = \frac{9}{4} = 2.25 \]
\[ x - 2 = 0 \quad \Rightarrow \quad x = 2 \]

### Step 2: Solve the second equation \(3y^2 - 8y + 4 = 0\)

1. **Identify coefficients**: Here, \(a = 3\), \(b = -8\), and \(c = 4\).
2. **Calculate the product \(ac\)**:
\[ ac = 3 \times 4 = 12 \]
3. **Find two numbers that multiply to \(ac\) (12) and add to \(b\) (-8)**: The numbers are \(-6\) and \(-2\).
4. **Rewrite the equation**:
\[ 3y^2 - 6y - 2y + 4 = 0 \]
5. **Factor by grouping**:
\[ (3y^2 - 6y) + (-2y + 4) = 0 \]
\[ 3y(y - 2) - 2(y - 2) = 0 \]
\[ (3y - 2)(y - 2) = 0 \]
6. **Set each factor to zero**:
\[ 3y - 2 = 0 \quad \Rightarrow \quad y = \frac{2}{3} \approx 0.67 \]
\[ y - 2 = 0 \quad \Rightarrow \quad y = 2 \]

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

1. **Values of \(x\)**: \(x = 2.25\) and \(x = 2\)
2. **Values of \(y\)**: \(y = \frac{2}{3} \approx 0.67\) and \(y = 2\)

### Step 4: Determine the relationships

- For \(x = 2.25\) and \(y = \frac{2}{3}\):
\[ 2.25 > 0.67 \quad \text{(so, \(x > y\))} \]
- For \(x = 2.25\) and \(y = 2\):
\[ 2.25 > 2 \quad \text{(so, \(x > y\))} \]
- For \(x = 2\) and \(y = \frac{2}{3}\):
\[ 2 > 0.67 \quad \text{(so, \(x > y\))} \]
- For \(x = 2\) and \(y = 2\):
\[ 2 = 2 \quad \text{(so, \(x = y\))} \]

### Conclusion

The relationship between \(x\) and \(y\) can be summarized as:
- \(x\) is either greater than or equal to \(y\).

Thus, the correct option is that \(x\) is greater than or equal to \(y\).

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