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  4. If the difference of the real roots of equation x2+4 x+(p/q)=0 is 1 (where p and q are coprime), then the value of (p+q) is

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Q. If the difference of the real roots of equation $x^2+4 x+\frac{p}{q}=0$ is 1 (where $p$ and $q$ are coprime), then the value of $(p+q)$ is

Complex Numbers and Quadratic Equations

Solution:

Given, $|\alpha-\beta|^2=1 \Rightarrow(\alpha+\beta)^2-4 \alpha \beta=1 \Rightarrow \frac{15}{4}=\alpha \beta$
$\text { So, } \frac{p}{q}=\frac{15}{4} $
$\therefore p=15, q=4$
$\text { Hence, } p+q=19 $