The zeroes of a polynomial $x^{2}+p x+q$ are twice the zeroes of the polynomial $4 x^{2}-5 x-6$. The

Question

The zeroes of a polynomial $x^{2}+p x+q$ are twice the zeroes of the polynomial $4 x^{2}-5 x-6$. The value of $p$ is :\

Professor PC Thomas Classes · Answer

(a) : Let the zeroes of the polynomial $x^{2}+p x+q$ be $\alpha$ and $\beta$. Then, $\alpha+\beta=-p, \alpha \beta=q$ Also, for the zeroes of the polynomial $p(x)=4 x^{2}-5 x-6$, put $p(x)=0$\ $\Rightarrow 4 x^{2}-5 x-6=0 \Rightarrow 4 x^{2}-8 x+3 x-6=0$\ $\Rightarrow 4 x(x-2)+3(x-2)=0 \Rightarrow(x-2)(4 x+3)=0$\ $\Rightarrow \quad x=2$ or $x=\frac{-3}{4}$ are the zeroes of $p(x)=0$.\ But twice the zeroes of $p(x)=$ zeroes of $x^{2}+p x+q$\ $\Rightarrow \alpha=4$ and $\beta=\frac{-3}{2} \quad 	herefore \alpha+\beta=4-\frac{3}{2}=\frac{5}{2}$\ $	herefore \quad$ The value of $p=\frac{-5}{2}$.\