Problem - 2599
Let $x_1$ and $x_2$ be the two roots of $2x^2 -7x -4=0$, compute the values of the following expressions using as many different ways as possible.
(1) $x_1^2 + x_2^2$
(2) $(x_1+1)(x_2+1)$
(3) $\mid x_1 - x_2 \mid$
(1) $x_1^2 + x_2^2 = (x_1+x_2)^2 - 2x_1x_2 = \Big(\frac{7}{2}\Big)^2 - 2\cdot (-2) = \boxed{\frac{65}{4}}$. (2) $(x_1+1)(x_2+1)=(x_1+x_2) + (x_1x_2)+1=\frac{7}{2} + (-2) + 1=\boxed{\frac{5}{2}}$ (3) $(x_1-x_2)^2 = (x_1+x_2)^2 -4x_1x_2 = \Big(\frac{7}{2}\Big)^2 - 4\cdot (-2)=\frac{81}{2}$. Therefore $|x_1-x_2|=\boxed{\frac{9}{2}}$.
Note: there are several different solutions. Please refer to the book Competition Algebra .