In each of these questions two equations numbered I and II are given. You have to solve both the equations and give answer
I. `5x^2-11x+2=0`
II. `3y^2-5y+2=0`

Options

A. `xley`
B. `xgty`
C. `xgey`
D. the relationship can’t be established

Answer

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

### Step 1: Solve Equation I
The first equation is:
\[ 5x^2 - 11x + 2 = 0 \]

We will use the quadratic formula to solve for \( x \):
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 5 \), \( b = -11 \), and \( c = 2 \).

#### Step 1.1: Calculate the Discriminant
First, calculate the discriminant \( D \):
\[ D = b^2 - 4ac = (-11)^2 - 4 \cdot 5 \cdot 2 \]
\[ D = 121 - 40 = 81 \]

#### Step 1.2: Apply the Quadratic Formula
Now, substitute the values into the quadratic formula:
\[ x = \frac{-(-11) \pm \sqrt{81}}{2 \cdot 5} \]
\[ x = \frac{11 \pm 9}{10} \]

This gives us two possible solutions:
1. \( x = \frac{11 + 9}{10} = \frac{20}{10} = 2 \)
2. \( x = \frac{11 - 9}{10} = \frac{2}{10} = 0.2 \)

### Step 2: Solve Equation II
The second equation is:
\[ 3y^2 - 5y + 2 = 0 \]

Again, we will use the quadratic formula:
\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 3 \), \( b = -5 \), and \( c = 2 \).

#### Step 2.1: Calculate the Discriminant
Calculate the discriminant \( D \):
\[ D = b^2 - 4ac = (-5)^2 - 4 \cdot 3 \cdot 2 \]
\[ D = 25 - 24 = 1 \]

#### Step 2.2: Apply the Quadratic Formula
Now, substitute the values into the quadratic formula:
\[ y = \frac{-(-5) \pm \sqrt{1}}{2 \cdot 3} \]
\[ y = \frac{5 \pm 1}{6} \]

This gives us two possible solutions:
1. \( y = \frac{5 + 1}{6} = \frac{6}{6} = 1 \)
2. \( y = \frac{5 - 1}{6} = \frac{4}{6} = \frac{2}{3} \approx 0.67 \)

### Summary of Solutions
From the first equation, we found:
- \( x = 2 \) and \( x = 0.2 \)

From the second equation, we found:
- \( y = 1 \) and \( y \approx 0.67 \)

### Step 3: Determine the Relationship
Now we compare the values:
- \( 0.2 < 0.67 \)
- \( 2 > 1 \)

Since \( 0.2 < 0.67 \) and \( 2 > 1 \), we can conclude that there is no direct relationship established between \( x \) and \( y \).

### Final Answer
The correct answer is that no relationship can be established between \( x \) and \( y \).

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