Find the value of X and Y (X और Y का मान ज्ञात कीजिए)
I. `x^(2)+9x+20=0`
II. `y^(2)+7y+12=0`

Options

A. `xley`
B. `xgty`
C. `x=y`
D. none of these(इनमें से कोई नहीं)

Answer

To solve the equations for \( X \) and \( Y \), we will follow these steps:

### Step 1: Solve for \( X \)
We have the equation:
\[ x^2 + 9x + 20 = 0 \]

To factor this quadratic equation, we need to find two numbers that multiply to \( 20 \) (the constant term) and add up to \( 9 \) (the coefficient of \( x \)).

The numbers that satisfy this condition are \( 4 \) and \( 5 \).

So we can rewrite the equation as:
\[ (x + 4)(x + 5) = 0 \]

### Step 2: Set each factor to zero
Now, we set each factor equal to zero:
1. \( x + 4 = 0 \) → \( x = -4 \)
2. \( x + 5 = 0 \) → \( x = -5 \)

Thus, the values of \( X \) are:
\[ X = -4 \quad \text{and} \quad X = -5 \]

### Step 3: Solve for \( Y \)
Next, we have the equation:
\[ y^2 + 7y + 12 = 0 \]

Again, we need to find two numbers that multiply to \( 12 \) and add up to \( 7 \).

The numbers that satisfy this condition are \( 3 \) and \( 4 \).

So we can rewrite the equation as:
\[ (y + 3)(y + 4) = 0 \]

### Step 4: Set each factor to zero
Now, we set each factor equal to zero:
1. \( y + 3 = 0 \) → \( y = -3 \)
2. \( y + 4 = 0 \) → \( y = -4 \)

Thus, the values of \( Y \) are:
\[ Y = -3 \quad \text{and} \quad Y = -4 \]

### Final Answer
The values of \( X \) and \( Y \) are:
- \( X = -4 \) or \( -5 \)
- \( Y = -3 \) or \( -4 \)

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