(i)` 7x^(2) + 16x - 15 = 0`
(ii) ` 5y^(2)+ 8y - 21 = 0`

Options

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

Explanation

To solve the given quadratic equations and find the relationship between the values of \( x \) and \( y \), we will follow these steps:

### Step 1: Solve the first equation \( 7x^2 + 16x - 15 = 0 \)

We will factor the quadratic equation. We need to find two numbers that multiply to \( 7 \times (-15) = -105 \) and add to \( 16 \).

The factors of \(-105\) that add up to \(16\) are \(21\) and \(-5\).

Now, we can rewrite the equation:

\[ 7x^2 + 21x - 5x - 15 = 0 \]

Next, we group the terms:

\[ (7x^2 + 21x) + (-5x - 15) = 0 \]

Factoring by grouping:

\[ 7x(x + 3) - 5(x + 3) = 0 \]

Now, we can factor out \((x + 3)\):

\[ (7x - 5)(x + 3) = 0 \]

Setting each factor to zero gives us:

1. \( 7x - 5 = 0 \) → \( x = \frac{5}{7} \)
2. \( x + 3 = 0 \) → \( x = -3 \)

So, the values of \( x \) are \( \frac{5}{7} \) and \( -3 \).

### Step 2: Solve the second equation \( 5y^2 + 8y - 21 = 0 \)

Similarly, we will factor this quadratic equation. We need to find two numbers that multiply to \( 5 \times (-21) = -105 \) and add to \( 8 \).

The factors of \(-105\) that add up to \(8\) are \(15\) and \(-7\).

Now, we can rewrite the equation:

\[ 5y^2 + 15y - 7y - 21 = 0 \]

Next, we group the terms:

\[ (5y^2 + 15y) + (-7y - 21) = 0 \]

Factoring by grouping:

\[ 5y(y + 3) - 7(y + 3) = 0 \]

Now, we can factor out \((y + 3)\):

\[ (5y - 7)(y + 3) = 0 \]

Setting each factor to zero gives us:

1. \( 5y - 7 = 0 \) → \( y = \frac{7}{5} \)
2. \( y + 3 = 0 \) → \( y = -3 \)

So, the values of \( y \) are \( \frac{7}{5} \) and \( -3 \).

### Step 3: Compare the values of \( x \) and \( y \)

Now we have the values:

- For \( x \): \( \frac{5}{7} \) and \( -3 \)
- For \( y \): \( \frac{7}{5} \) and \( -3 \)

We can see that both equations have a common value of \( -3 \).

### Conclusion

The relationship between \( x \) and \( y \) is that they both can be equal when \( x = y = -3 \). Therefore, we conclude that:

\[ x = y \text{ when } x = -3 \text{ and } y = -3. \]