In each questions two equations numbered I. and II. are given. You have to solve both the equations and mark appropriate answer
I. ` 2 x^(2) - 17 x + 36 = 0 `
II. ` 3y ^(2) - 22y + 40 = 0 `

Options

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

Answer

To solve the given equations step by step, we will start with each equation separately.

### Step 1: Solve Equation I
The first equation is:
\[ 2x^2 - 17x + 36 = 0 \]

**Step 1.1: Factor the quadratic equation**
To factor the quadratic equation, we need two numbers that multiply to \(2 \times 36 = 72\) and add up to \(-17\). The numbers that satisfy this condition are \(-8\) and \(-9\).

We can rewrite the equation as:
\[ 2x^2 - 8x - 9x + 36 = 0 \]

**Step 1.2: Group the terms**
Now, we group the terms:
\[ (2x^2 - 8x) + (-9x + 36) = 0 \]

**Step 1.3: Factor by grouping**
Factoring out the common terms:
\[ 2x(x - 4) - 9(x - 4) = 0 \]

Now we can factor out \((x - 4)\):
\[ (x - 4)(2x - 9) = 0 \]

**Step 1.4: Solve for x**
Setting each factor to zero gives us:
1. \( x - 4 = 0 \) → \( x = 4 \)
2. \( 2x - 9 = 0 \) → \( 2x = 9 \) → \( x = \frac{9}{2} = 4.5 \)

### Step 2: Solve Equation II
The second equation is:
\[ 3y^2 - 22y + 40 = 0 \]

**Step 2.1: Factor the quadratic equation**
We need two numbers that multiply to \(3 \times 40 = 120\) and add up to \(-22\). The numbers that satisfy this condition are \(-12\) and \(-10\).

We can rewrite the equation as:
\[ 3y^2 - 12y - 10y + 40 = 0 \]

**Step 2.2: Group the terms**
Now, we group the terms:
\[ (3y^2 - 12y) + (-10y + 40) = 0 \]

**Step 2.3: Factor by grouping**
Factoring out the common terms:
\[ 3y(y - 4) - 10(y - 4) = 0 \]

Now we can factor out \((y - 4)\):
\[ (y - 4)(3y - 10) = 0 \]

**Step 2.4: Solve for y**
Setting each factor to zero gives us:
1. \( y - 4 = 0 \) → \( y = 4 \)
2. \( 3y - 10 = 0 \) → \( 3y = 10 \) → \( y = \frac{10}{3} \approx 3.33 \)

### Step 3: Compare the values of x and y
We have found the values:
- \( x = 4 \) or \( x = 4.5 \)
- \( y = 4 \) or \( y \approx 3.33 \)

**Step 3.1: Compare \( x \) and \( y \)**
1. Comparing \( x = 4 \) and \( y = 4 \):
- \( x = y \)
2. Comparing \( x = 4.5 \) and \( y = 4 \):
- \( x > y \)
3. Comparing \( x = 4.5 \) and \( y \approx 3.33 \):
- \( x > y \)

### Conclusion
From the comparisons, we can conclude:
- \( x \) is greater than or equal to \( y \).

Thus, the final answer is:
\[ x \geq y \]

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