Find the values of `m` for which the expression `2x^2+m x y+3y^2-5y-2` can be resolved into two rational linear factors.

Options

A. 3
B. 5
C. 7
D. 9

Answer

To find the values of \( m \) for which the expression \( 2x^2 + mxy + 3y^2 - 5y - 2 \) can be resolved into two rational linear factors, we will follow these steps:

### Step 1: Identify the expression
We start with the expression:
\[ 2x^2 + mxy + 3y^2 - 5y - 2 \]

### Step 2: Rewrite the expression
We can rearrange the expression in the standard form of a quadratic in \( y \):
\[ 3y^2 + (mx - 5)y + (2x^2 - 2) \]

### Step 3: Identify coefficients
Now, we identify the coefficients \( a \), \( b \), and \( c \) from the quadratic equation \( ay^2 + by + c = 0 \):
- \( a = 3 \)
- \( b = mx - 5 \)
- \( c = 2x^2 - 2 \)

### Step 4: Use the condition for rational factors
For the quadratic to be factorable into rational linear factors, the discriminant must be zero. The discriminant \( D \) is given by:
\[ D = b^2 - 4ac \]
Substituting our coefficients:
\[ D = (mx - 5)^2 - 4 \cdot 3 \cdot (2x^2 - 2) \]

### Step 5: Simplify the discriminant
Now we simplify the discriminant:
\[ D = (mx - 5)^2 - 12(2x^2 - 2) \]
\[ D = (mx - 5)^2 - 24x^2 + 24 \]

### Step 6: Expand and combine like terms
Expanding \( (mx - 5)^2 \):
\[ D = m^2x^2 - 10mx + 25 - 24x^2 + 24 \]
Combining like terms:
\[ D = (m^2 - 24)x^2 - 10mx + 49 \]

### Step 7: Set the discriminant to zero
For the quadratic in \( x \) to have rational roots, we set the discriminant \( D \) to zero:
\[ (m^2 - 24)x^2 - 10mx + 49 = 0 \]
The discriminant of this quadratic must also be zero:
\[ (-10m)^2 - 4(m^2 - 24)(49) = 0 \]

### Step 8: Solve the equation
Calculating the discriminant:
\[ 100m^2 - 4(m^2 - 24)(49) = 0 \]
Expanding:
\[ 100m^2 - 196m^2 + 2352 = 0 \]
Combining terms:
\[ -96m^2 + 2352 = 0 \]
Solving for \( m^2 \):
\[ 96m^2 = 2352 \]
\[ m^2 = \frac{2352}{96} = 24.5 \]

### Step 9: Find \( m \)
Taking the square root:
\[ m = \pm \sqrt{24.5} \]

### Final Answer
Thus, the values of \( m \) for which the expression can be resolved into two rational linear factors are:
\[ m = \pm 7 \]