If the quadratic equation ` 3x^(2) + ax+1=0 and 2x^(2) + bx+1=0` have a common root , then the value of the expression ` 5ab-2a^(2)-3b^(2)` is

Options

A. 0
B. 1
C. `-1`
D. None of these

Answer

To solve the problem, we need to find the value of the expression \( 5ab - 2a^2 - 3b^2 \) given that the quadratic equations \( 3x^2 + ax + 1 = 0 \) and \( 2x^2 + bx + 1 = 0 \) have a common root.

### Step 1: Identify the common root
Let the common root be \( \alpha \). Since \( \alpha \) is a root of both equations, it satisfies both:

1. \( 3\alpha^2 + a\alpha + 1 = 0 \) (Equation 1)
2. \( 2\alpha^2 + b\alpha + 1 = 0 \) (Equation 2)

### Step 2: Multiply the equations to eliminate \( \alpha^2 \)
To eliminate \( \alpha^2 \), we can multiply Equation 1 by 2 and Equation 2 by 3:

- Multiplying Equation 1 by 2:
\[ 6\alpha^2 + 2a\alpha + 2 = 0 \quad (Equation 3) \]

- Multiplying Equation 2 by 3:
\[ 6\alpha^2 + 3b\alpha + 3 = 0 \quad (Equation 4) \]

### Step 3: Set the two equations equal
Now, we can set Equation 3 equal to Equation 4 since both equal \( 6\alpha^2 \):

\[ 2a\alpha + 2 = 3b\alpha + 3 \]

### Step 4: Rearrange the equation
Rearranging gives us:

\[ 2a\alpha - 3b\alpha + 2 - 3 = 0 \]
\[ (2a - 3b)\alpha - 1 = 0 \]

### Step 5: Solve for \( \alpha \)
From the equation above, we can express \( \alpha \):

\[ (2a - 3b)\alpha = 1 \]
\[ \alpha = \frac{1}{2a - 3b} \quad (if \; 2a - 3b \neq 0) \]

### Step 6: Substitute \( \alpha \) back into one of the original equations
We can substitute \( \alpha \) back into either Equation 1 or Equation 2. Let's use Equation 1:

\[ 3\left(\frac{1}{2a - 3b}\right)^2 + a\left(\frac{1}{2a - 3b}\right) + 1 = 0 \]

### Step 7: Simplify the equation
Substituting and simplifying gives:

\[ \frac{3}{(2a - 3b)^2} + \frac{a}{2a - 3b} + 1 = 0 \]

Multiplying through by \( (2a - 3b)^2 \) to eliminate the denominator:

\[ 3 + a(2a - 3b) + (2a - 3b)^2 = 0 \]

### Step 8: Expand and collect like terms
Expanding gives:

\[ 3 + 2a^2 - 3ab + 4a^2 - 12ab + 9b^2 = 0 \]
\[ 6a^2 - 15ab + 9b^2 + 3 = 0 \]

### Step 9: Rearranging the equation
Rearranging gives us:

\[ 6a^2 - 15ab + 9b^2 = -3 \]

### Step 10: Find the value of the expression
We need to find \( 5ab - 2a^2 - 3b^2 \). From our derived equation, we can express \( 5ab - 2a^2 - 3b^2 \):

\[ 5ab - 2a^2 - 3b^2 = 1 \]

### Final Answer
Thus, the value of \( 5ab - 2a^2 - 3b^2 \) is \( \boxed{1} \).