Given below are two equations in each question, which you have to solve and give answer
I. ` x^(2) - 7 x + 10 = 0`
II. ` y^(2) - 14 y + 45 = 0`

Options

A. If ` x gt y`
B. If ` x ge y `
C. If ` y gt x`
D. If ` y ge x `

Answer

To solve the given equations step by step, we will first tackle each equation separately.

### Step 1: Solve the first equation \( x^2 - 7x + 10 = 0 \)

1. **Factor the quadratic equation**: We need to express the equation in a factored form. We look for two numbers that multiply to \(10\) (the constant term) and add to \(-7\) (the coefficient of \(x\)).
- The numbers are \(-5\) and \(-2\).

Thus, we can factor the equation as:
\[ (x - 5)(x - 2) = 0 \]

2. **Set each factor to zero**:
- \(x - 5 = 0 \Rightarrow x = 5\)
- \(x - 2 = 0 \Rightarrow x = 2\)

So the solutions for the first equation are:
\[ x = 5 \quad \text{and} \quad x = 2 \]

### Step 2: Solve the second equation \( y^2 - 14y + 45 = 0 \)

1. **Factor the quadratic equation**: Similarly, we need to find two numbers that multiply to \(45\) and add to \(-14\).
- The numbers are \(-9\) and \(-5\).

Thus, we can factor the equation as:
\[ (y - 9)(y - 5) = 0 \]

2. **Set each factor to zero**:
- \(y - 9 = 0 \Rightarrow y = 9\)
- \(y - 5 = 0 \Rightarrow y = 5\)

So the solutions for the second equation are:
\[ y = 9 \quad \text{and} \quad y = 5 \]

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

Now we have the solutions:
- For \(x\): \(2\) and \(5\)
- For \(y\): \(9\) and \(5\)

We will compare these values:
1. If we take \(x = 2\):
- Compare with \(y = 9\): \(2 < 9\)
- Compare with \(y = 5\): \(2 < 5\)

2. If we take \(x = 5\):
- Compare with \(y = 9\): \(5 < 9\)
- Compare with \(y = 5\): \(5 = 5\)

### Conclusion

From the comparisons, we find that:
- \(x\) can be less than \(y\) (when \(x = 2\) and \(y = 5\) or \(y = 9\)).
- \(x\) can also be equal to \(y\) (when \(x = 5\) and \(y = 5\)).

Thus, we can conclude that:
\[ x \leq y \]

### Final Answer
The relation between \(x\) and \(y\) is \(x \leq y\).

---