18. Let $p, q, r, s$ be the four roots of the equation $2(10 x+13)^{2}(5 x+8)(x+1)=1$. If $p q+r s$ is real, find the value of this real number.
(2 marks)
設 $p 、 q 、 r 、 s$ 為方程 $2(10 x+13)^{2}(5 x+8)(x+1)=1$ 的四個根。若 $p q+r s$ 是實數, 求此實數的值。
(2 分)
18. $\frac{329}{100}$
18. Upon expansion, we get $2\left(100 x^{2}+260 x+169\right)\left(5 x^{2}+13 x+8\right)=1$. If we let $u=5 x^{2}+13 x+8$, the equation becomes $2(20 u+9) u=1$, which gives $u=\frac{1}{20}$ or $u=-\frac{1}{2}$, i.e. $5 x^{2}+13 x+\frac{159}{20}=0$ or $5 x^{2}+13 x+\frac{17}{2}=0$. The former gives two real roots with product $\frac{159}{100}$, and the latter gives two complex roots with product $\frac{17}{10}$. The two roots in the same equation must be paired up in order for $p q+r s$ to be real. It follows that the answer is $\frac{159}{100}+\frac{17}{10}=\frac{329}{100}$.