In each of the these question two equation (I) and (II) are given. You have to solve both the equtions and give answer accordingly
I `21x^2+10x+1=0`
II `24y^2+26y+5=0`

Options

A. `xgty`
B. `xgey`
C. `xley`
D. no relationship can be established b/w x and y

Answer

To solve the equations given in the question, we will follow these steps:

### Step 1: Solve the first equation \(21x^2 + 10x + 1 = 0\)

1. **Identify coefficients**: Here, \(a = 21\), \(b = 10\), and \(c = 1\).
2. **Use the quadratic formula**: The roots of the quadratic equation \(ax^2 + bx + c = 0\) can be found using the formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
3. **Calculate the discriminant**:
\[ D = b^2 - 4ac = 10^2 - 4 \cdot 21 \cdot 1 = 100 - 84 = 16 \]
4. **Calculate the roots**:
\[ x = \frac{-10 \pm \sqrt{16}}{2 \cdot 21} = \frac{-10 \pm 4}{42} \]
- For \(x_1\):
\[ x_1 = \frac{-10 + 4}{42} = \frac{-6}{42} = -\frac{1}{7} \]
- For \(x_2\):
\[ x_2 = \frac{-10 - 4}{42} = \frac{-14}{42} = -\frac{1}{3} \]

### Step 2: Solve the second equation \(24y^2 + 26y + 5 = 0\)

1. **Identify coefficients**: Here, \(a = 24\), \(b = 26\), and \(c = 5\).
2. **Use the quadratic formula**:
\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
3. **Calculate the discriminant**:
\[ D = b^2 - 4ac = 26^2 - 4 \cdot 24 \cdot 5 = 676 - 480 = 196 \]
4. **Calculate the roots**:
\[ y = \frac{-26 \pm \sqrt{196}}{2 \cdot 24} = \frac{-26 \pm 14}{48} \]
- For \(y_1\):
\[ y_1 = \frac{-26 + 14}{48} = \frac{-12}{48} = -\frac{1}{4} \]
- For \(y_2\):
\[ y_2 = \frac{-26 - 14}{48} = \frac{-40}{48} = -\frac{5}{6} \]

### Step 3: Summarize the results

- The solutions for \(x\) are:
- \(x_1 = -\frac{1}{7} \approx -0.1428\)
- \(x_2 = -\frac{1}{3} \approx -0.3333\)

- The solutions for \(y\) are:
- \(y_1 = -\frac{1}{4} = -0.25\)
- \(y_2 = -\frac{5}{6} \approx -0.8333\)

### Step 4: Analyze the relationship between \(x\) and \(y\)

- Comparing the values:
- \(x_1 \approx -0.1428\) and \(y_2 \approx -0.8333\): \(x_1 > y_2\)
- \(x_2 \approx -0.3333\) and \(y_1 = -0.25\): \(y_1 > x_2\)

Since there is no consistent relationship between the values of \(x\) and \(y\), we conclude that there is no relationship that can be established.

### Final Answer
The correct answer is: **No relationship can be established between \(x\) and \(y\)**.

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