In each of these questions, two equations I. and II. are given. You have to solve both the equations and give answer
I. ` x^(2) - 9 x + 20 = 0 `
II. ` 2 y ^(2) - 12 y + 18 = 0 `

Options

A. If ` x gt y `
B. If ` x ge y`
C. If x = y or no relation can be established between x and y
D. If `y gt x `

Answer

To solve the given equations step by step, let's start with the first equation.

### Step 1: Solve Equation I
The first equation is:
\[ x^2 - 9x + 20 = 0 \]

This is a quadratic equation, and we can factor it. We need to find two numbers that multiply to \(20\) (the constant term) and add up to \(-9\) (the coefficient of \(x\)).

The numbers that satisfy this condition are \(-4\) and \(-5\) because:
- \(-4 \times -5 = 20\)
- \(-4 + -5 = -9\)

So we can factor the equation as:
\[ (x - 4)(x - 5) = 0 \]

### Step 2: Find the values of \(x\)
Now, we set each factor to zero:
1. \(x - 4 = 0\) → \(x = 4\)
2. \(x - 5 = 0\) → \(x = 5\)

Thus, the solutions for \(x\) are:
\[ x = 4 \quad \text{and} \quad x = 5 \]

### Step 3: Solve Equation II
The second equation is:
\[ 2y^2 - 12y + 18 = 0 \]

To simplify, we can divide the entire equation by \(2\):
\[ y^2 - 6y + 9 = 0 \]

Next, we can factor this equation. We need two numbers that multiply to \(9\) and add up to \(-6\). The numbers are \(-3\) and \(-3\):
- \(-3 \times -3 = 9\)
- \(-3 + -3 = -6\)

So we can factor the equation as:
\[ (y - 3)(y - 3) = 0 \]
or simply:
\[ (y - 3)^2 = 0 \]

### Step 4: Find the value of \(y\)
Setting the factor to zero gives us:
\[ y - 3 = 0 \]
Thus, the solution for \(y\) is:
\[ y = 3 \]

### Summary of Solutions
The solutions we found are:
- \(x = 4\) and \(x = 5\)
- \(y = 3\)

### Step 5: Compare \(x\) and \(y\)
Now, we compare the values of \(x\) and \(y\):
- For \(x = 4\): \(4 > 3\)
- For \(x = 5\): \(5 > 3\)

Thus, we conclude that:
\[ x > y \]

### Final Answer
The relationship between \(x\) and \(y\) is:
\[ x \text{ is greater than } y \]

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