In each question two equations numbered (I) and (II) are given . You have to solve both the equations and mark appropriate answer
I. ` 17 x^(2) - 14 x - 83 = - 80`
II. ` y^(2) = 2y + 35`

Options

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

Answer

To solve the given equations step by step, we will follow these procedures:

### Step 1: Solve Equation I
The first equation is:
\[ 17x^2 - 14x - 83 = -80 \]

We can rearrange this equation to set it equal to zero:
\[ 17x^2 - 14x - 83 + 80 = 0 \]
\[ 17x^2 - 14x - 3 = 0 \]

### Step 2: Factor the Quadratic Equation
We need to factor the quadratic equation \( 17x^2 - 14x - 3 = 0 \). To do this, we look for two numbers that multiply to \( 17 \times (-3) = -51 \) and add to \( -14 \).

The factors of \(-51\) that add up to \(-14\) are \(-17\) and \(3\).

Now we can rewrite the equation:
\[ 17x^2 - 17x + 3x - 3 = 0 \]

### Step 3: Grouping
Next, we group the terms:
\[ (17x^2 - 17x) + (3x - 3) = 0 \]
Factoring out the common factors:
\[ 17x(x - 1) + 3(x - 1) = 0 \]

### Step 4: Factor Out the Common Term
Now we can factor out \( (x - 1) \):
\[ (x - 1)(17x + 3) = 0 \]

### Step 5: Solve for x
Setting each factor to zero gives us:
1. \( x - 1 = 0 \) → \( x = 1 \)
2. \( 17x + 3 = 0 \) → \( 17x = -3 \) → \( x = -\frac{3}{17} \approx -0.176 \)

### Step 6: Solve Equation II
The second equation is:
\[ y^2 = 2y + 35 \]

Rearranging gives us:
\[ y^2 - 2y - 35 = 0 \]

### Step 7: Factor the Quadratic Equation
Now we need to factor \( y^2 - 2y - 35 = 0 \). We look for two numbers that multiply to \(-35\) and add to \(-2\).

The factors of \(-35\) that add up to \(-2\) are \(-7\) and \(5\).

Rewriting the equation:
\[ (y - 7)(y + 5) = 0 \]

### Step 8: Solve for y
Setting each factor to zero gives us:
1. \( y - 7 = 0 \) → \( y = 7 \)
2. \( y + 5 = 0 \) → \( y = -5 \)

### Step 9: Compare Values of x and y
Now we have the values:
- For \( x \): \( 1 \) and \( -\frac{3}{17} \) (approximately \( -0.176 \))
- For \( y \): \( 7 \) and \( -5 \)

### Step 10: Establish Relationships
Now we compare the values:
1. \( 1 < 7 \) (True)
2. \( 1 > -5 \) (True)
3. \( -\frac{3}{17} < 7 \) (True)
4. \( -\frac{3}{17} > -5 \) (True)

### Conclusion
From the comparisons, we can see that:
- \( x \) can be both less than and greater than \( y \) depending on the values chosen.
- Therefore, we conclude that no specific relation can be established between \( x \) and \( y \).

### Final Answer
The answer is: **No relation can be established between x and y.**

---