Solve the given quadratic equations and mark the correct options based on your answer
I. ` x^(2) + 9 x = 25 x - 63`
II. ` 4 y^(2) - 34 y + 72 = 0`

Options

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

Answer

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

### Step 1: Solve the first quadratic equation
The first equation is:
\[ x^2 + 9x = 25x - 63 \]

**Rearranging the equation:**
Move all terms to one side:
\[ x^2 + 9x - 25x + 63 = 0 \]
This simplifies to:
\[ x^2 - 16x + 63 = 0 \]

**Step 2: Factor the quadratic equation**
We need to factor the quadratic equation \( x^2 - 16x + 63 \). We are looking for two numbers that multiply to \( 63 \) and add up to \( -16 \). The numbers \( -9 \) and \( -7 \) fit this requirement.

Thus, we can factor the equation as:
\[ (x - 9)(x - 7) = 0 \]

**Step 3: Find the values of x**
Setting each factor to zero gives us:
1. \( x - 9 = 0 \) → \( x = 9 \)
2. \( x - 7 = 0 \) → \( x = 7 \)

So, the solutions for \( x \) are \( x = 9 \) and \( x = 7 \).

### Step 4: Solve the second quadratic equation
The second equation is:
\[ 4y^2 - 34y + 72 = 0 \]

**Step 5: Simplify the equation**
We can divide the entire equation by \( 2 \) to simplify it:
\[ 2y^2 - 17y + 36 = 0 \]

**Step 6: Factor the quadratic equation**
Now we need to factor \( 2y^2 - 17y + 36 \). We are looking for two numbers that multiply to \( 72 \) (the product of \( 2 \) and \( 36 \)) and add up to \( -17 \). The numbers \( -9 \) and \( -8 \) fit this requirement.

Thus, we can factor the equation as:
\[ (2y - 9)(y - 4) = 0 \]

**Step 7: Find the values of y**
Setting each factor to zero gives us:
1. \( 2y - 9 = 0 \) → \( y = 4.5 \)
2. \( y - 4 = 0 \) → \( y = 4 \)

So, the solutions for \( y \) are \( y = 4.5 \) and \( y = 4 \).

### Step 8: Compare the values of x and y
Now we have:
- Values of \( x \): \( 9, 7 \)
- Values of \( y \): \( 4.5, 4 \)

**Comparison:**
- \( 9 > 4.5 \)
- \( 9 > 4 \)
- \( 7 > 4.5 \)
- \( 7 > 4 \)

Thus, \( x \) is greater than \( y \).

### Conclusion
The correct option based on the comparison is option D: \( x > y \).

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