Question

If $\alpha,\beta$ are roots of $2x^2-x-3\lambda=0$ and $\alpha,\gamma$ are roots of $2x^2+9x+2\lambda=0$, find equation of roots $2\alpha+\beta$ and $\beta+\gamma$.

• A. $x^2-x-2=0$
• B. $x^2+x-2=0$
• C. $x^2-3x+2=0$
• D. $x^2+3x+2=0$

Hint:

Use symmetric relations from Vieta directly.

Answer 1

The Correct Option is A

Step 1: From first equation: \[ \alpha+\beta=\frac{1}{2},\quad \alpha\beta=-\frac{3\lambda}{2} \]
Step 2: From second: \[ \alpha+\gamma=-\frac{9}{2},\quad \alpha\gamma=\lambda \]
Step 3: Express $\beta,\gamma$ in terms of $\alpha$: \[ \beta=\frac{1}{2}-\alpha,\quad \gamma=-\frac{9}{2}-\alpha \]
Step 4: Compute roots: \[ 2\alpha+\beta = \frac{1}{2}+\alpha \] \[ \beta+\gamma = -5-2\alpha \]
Step 5: Sum of roots: \[ S=-\frac{9}{2} \]
Step 6: Product: \[ P=2 \]
Step 7: Equation: \[ x^2-x-2=0 \]