I. ` x^(2) - 3x + 2 = 0" "II. 2y^(2) - 7y+6 = 0`

Options

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

Explanation

To solve the equations \( x^2 - 3x + 2 = 0 \) and \( 2y^2 - 7y + 6 = 0 \), we will find the values of \( x \) and \( y \) step by step.

### Step 1: Solve the first equation for \( x \)

The first equation is:
\[ x^2 - 3x + 2 = 0 \]

To factor this quadratic equation, we look for two numbers that multiply to \( 2 \) (the constant term) and add up to \( -3 \) (the coefficient of \( x \)). The numbers that satisfy this are \( -1 \) and \( -2 \).

Thus, we can factor the equation as:
\[ (x - 1)(x - 2) = 0 \]

### Step 2: Find the values of \( x \)

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

So, the solutions for \( x \) are:
\[ x = 1 \quad \text{and} \quad x = 2 \]

### Step 3: Solve the second equation for \( y \)

The second equation is:
\[ 2y^2 - 7y + 6 = 0 \]

To factor this quadratic equation, we first multiply the coefficient of \( y^2 \) (which is \( 2 \)) by the constant term \( 6 \) to get \( 12 \). We need two numbers that multiply to \( 12 \) and add up to \( -7 \). The numbers that work are \( -3 \) and \( -4 \).

We can rewrite the equation as:
\[ 2y^2 - 3y - 4y + 6 = 0 \]

Now, we can group the terms:
\[ y(2y - 3) - 2(2y - 3) = 0 \]

Factoring out \( (2y - 3) \):
\[ (2y - 3)(y - 2) = 0 \]

### Step 4: Find the values of \( y \)

Setting each factor equal to zero gives us:
1. \( 2y - 3 = 0 \) → \( y = \frac{3}{2} \)
2. \( y - 2 = 0 \) → \( y = 2 \)

So, the solutions for \( y \) are:
\[ y = \frac{3}{2} \quad \text{and} \quad y = 2 \]

### Step 5: Establish the relationship between \( x \) and \( y \)

Now we have:
- \( x = 1 \) or \( x = 2 \)
- \( y = \frac{3}{2} \) or \( y = 2 \)

To establish a relationship, we can compare the values:
- For \( x = 1 \), \( y = \frac{3}{2} \) (which is greater than \( x \)) and \( y = 2 \) (which is also greater than \( x \)).
- For \( x = 2 \), \( y = 2 \) (which is equal to \( x \)).

Thus, the relationship can be summarized as:
- \( y \geq x \)