Question

If the difference between the roots of $x^2+ax-b=0$ is equal to the difference between the roots of $x^2-px+q=0$, then $a^2-p^2$ in terms of $b$ and $q$ is equal

• A. $-4\left(b+q\right)$
• B. $4\left(b+q\right)$
• C. $4\left(b-q\right)$
• D. $4\left(q-b\right)$

Answer 1

The correct option is A

$-4\left(b+q\right)$

Explanation for the correct option.

Step 1: Find the relation between roots

Given that the difference between the roots of $x^2+ax-b=0$ is equal to the difference between the roots $x^2-px+q=0$.

Let $\alpha ,\beta$ are the roots of the equation $x^2+ax-b=0$ then:
The sum of the roots is: $\alpha +\beta =-a$

The product of the roots is: $\alpha \beta =-b$
Let, $\gamma ,\delta$ are the roots of the equation $x^2-px+q=0$ then,

The sum of the roots is: $\gamma +\delta =p$

The product of the roots is: $\gamma \delta =q$

Step 2: Find the value of the expression

According to the question:

The difference between roots is equal i.e; $\alpha -\beta =\gamma -\delta$.
$$\begin{gathered} \Rightarrow (\alpha -\beta {)}^2=(\gamma -\delta {)}^2 \\ \Rightarrow (\alpha +\beta {)}^2-4\alpha \beta =(\gamma +\delta {)}^2-4\gamma \delta \\ \Rightarrow a^2+4b=p^2-4q \\ \Rightarrow a^2-p^2=-4(b+q) \end{gathered}$$
Hence, option A is correct.