In each of these questions, two equations (I) and (II) are given . You have to solve both the equations and give answer
I.` x^(2) - 19 x + 84 = 0`
II. ` y^(2) - 25 y + 156 = 0 `

Options

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

Answer

To solve the given equations step by step, we will first tackle each equation separately.

### Step 1: Solve Equation I
The first equation is:
\[ x^2 - 19x + 84 = 0 \]

To solve this quadratic equation, we can factor it. We need to find two numbers that multiply to \( 84 \) (the constant term) and add up to \( -19 \) (the coefficient of \( x \)).

The numbers that satisfy these conditions are \( -12 \) and \( -7 \) because:
- \( -12 \times -7 = 84 \)
- \( -12 + -7 = -19 \)

Now, we can rewrite the equation as:
\[ (x - 12)(x - 7) = 0 \]

Setting each factor to zero gives us the solutions:
1. \( x - 12 = 0 \) → \( x = 12 \)
2. \( x - 7 = 0 \) → \( x = 7 \)

So, the solutions for \( x \) are:
\[ x = 12 \quad \text{and} \quad x = 7 \]

### Step 2: Solve Equation II
The second equation is:
\[ y^2 - 25y + 156 = 0 \]

Similarly, we will factor this quadratic equation. We need to find two numbers that multiply to \( 156 \) and add up to \( -25 \).

The numbers that satisfy these conditions are \( -13 \) and \( -12 \) because:
- \( -13 \times -12 = 156 \)
- \( -13 + -12 = -25 \)

Now, we can rewrite the equation as:
\[ (y - 13)(y - 12) = 0 \]

Setting each factor to zero gives us the solutions:
1. \( y - 13 = 0 \) → \( y = 13 \)
2. \( y - 12 = 0 \) → \( y = 12 \)

So, the solutions for \( y \) are:
\[ y = 13 \quad \text{and} \quad y = 12 \]

### Step 3: Compare Values of \( x \) and \( y \)
Now we have the values:
- For \( x \): \( 12 \) and \( 7 \)
- For \( y \): \( 13 \) and \( 12 \)

We need to compare these values:
- \( 7 < 12 \)
- \( 12 = 12 \)
- \( 12 < 13 \)

### Conclusion
From the comparisons, we can conclude that:
- The maximum value of \( x \) is \( 12 \) which is equal to the minimum value of \( y \) which is also \( 12 \).
- The minimum value of \( x \) is \( 7 \) which is less than both values of \( y \).

Thus, we can say:
\[ x \leq y \]

### Final Answer
The relationship between \( x \) and \( y \) is \( x \leq y \).

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