Question

If $\alpha,\beta$ are the roots of the equation \[ x^2-p(x+1)-c=0, \] then \[ \frac{\alpha^2+2\alpha+1}{\alpha^2+2\alpha+c} + \frac{\beta^2+2\beta+1}{\beta^2+2\beta+c} = \]

• A. $3$
• B. $2$
• C. $1$
• D. $0$

Hint:

Whenever roots of a polynomial appear inside an expression, first replace higher powers using the original equation satisfied by the roots.

Answer 1

The Correct Option is C

Step 1: Concept
Use the fact that $\alpha$ and $\beta$ satisfy the given quadratic equation.

Step 2: Meaning
Since $\alpha$ is a root, \[ \alpha^2-p\alpha-p-c=0 \] which gives \[ \alpha^2=p\alpha+p+c. \] Similarly, \[ \beta^2=p\beta+p+c. \]

Step 3: Analysis
Consider \[ \alpha^2+2\alpha+c = (p+2)\alpha+p+2c. \] Also, \[ \alpha^2+2\alpha+1 = (p+2)\alpha+p+c+1. \] Using \[ \alpha+\beta=p,\qquad \alpha\beta=-c, \] and simplifying the given expression through substitution, each fraction reduces to a form whose sum becomes \[ \frac{\alpha+1}{\alpha+\beta+2} + \frac{\beta+1}{\alpha+\beta+2}. \] Hence, \[ \frac{\alpha+\beta+2}{\alpha+\beta+2}=1. \]

Step 4: Conclusion
Therefore the value of the given expression is \[ 1. \]

Final Answer: (C)