Find the value of X and Y (X और Y का मान ज्ञात कीजिए)
I. `x^(2)+13x+42=0`
II. `y^(2)+16y+63=0`

Options

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

Answer

To solve the equations given in the problem, we will break it down step by step.

### Step 1: Solve for X
We start with the equation:
\[ x^2 + 13x + 42 = 0 \]

To factor this equation, we need two numbers that multiply to 42 (the constant term) and add up to 13 (the coefficient of x).

The numbers that satisfy this condition are 6 and 7 because:
\[ 6 \times 7 = 42 \]
\[ 6 + 7 = 13 \]

Now we can rewrite the quadratic equation as:
\[ x^2 + 6x + 7x + 42 = 0 \]

### Step 2: Factor the equation
Next, we group the terms:
\[ (x^2 + 6x) + (7x + 42) = 0 \]

Now, we can factor by grouping:
\[ x(x + 6) + 7(x + 6) = 0 \]

This gives us:
\[ (x + 6)(x + 7) = 0 \]

### Step 3: Solve for X
Now we can set each factor equal to zero:
1. \( x + 6 = 0 \) → \( x = -6 \)
2. \( x + 7 = 0 \) → \( x = -7 \)

Thus, the values of X are:
\[ x = -6 \text{ or } x = -7 \]

### Step 4: Solve for Y
Now we move on to the second equation:
\[ y^2 + 16y + 63 = 0 \]

Again, we need two numbers that multiply to 63 and add up to 16. The numbers that satisfy this condition are 7 and 9 because:
\[ 7 \times 9 = 63 \]
\[ 7 + 9 = 16 \]

We can rewrite the quadratic equation as:
\[ y^2 + 7y + 9y + 63 = 0 \]

### Step 5: Factor the equation
Now, we group the terms:
\[ (y^2 + 7y) + (9y + 63) = 0 \]

Factoring by grouping, we get:
\[ y(y + 7) + 9(y + 7) = 0 \]

This gives us:
\[ (y + 7)(y + 9) = 0 \]

### Step 6: Solve for Y
Now we can set each factor equal to zero:
1. \( y + 7 = 0 \) → \( y = -7 \)
2. \( y + 9 = 0 \) → \( y = -9 \)

Thus, the values of Y are:
\[ y = -7 \text{ or } y = -9 \]

### Final Answer
The values are:
- For X: \( x = -6 \) or \( x = -7 \)
- For Y: \( y = -7 \) or \( y = -9 \)

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