Question

2．已知方程2x^a+b-x^a-b-ab=0是关于x的一元二次方程，求a、b的值．

Answer 1

分析 本题根据一元二次方程的定义求解．分5种情况分别求解即可．

解答 解：∵2x^a+b-x^a-b+ab=0是关于x的一元二次方程，
∴①$\left\{\begin{array}{l}{a+b=2}\\{a-b=1}\end{array}\right.$，解得$\left\{\begin{array}{l}{a=\frac{3}{2}}\\{b=\frac{1}{2}}\end{array}\right.$；
②$\left\{\begin{array}{l}{a+b=2}\\{a-b=0}\end{array}\right.$，解得$\left\{\begin{array}{l}{a=1}\\{b=1}\end{array}\right.$；
③$\left\{\begin{array}{l}{a+b=1}\\{a-b=2}\end{array}\right.$，解得$\left\{\begin{array}{l}{a=\frac{3}{2}}\\{b=-\frac{1}{2}}\end{array}\right.$；
④$\left\{\begin{array}{l}{a+b=0}\\{a-b=2}\end{array}\right.$，解得$\left\{\begin{array}{l}{a=1}\\{b=-1}\end{array}\right.$；
⑤$\left\{\begin{array}{l}{a+b=2}\\{a-b=2}\end{array}\right.$，解得$\left\{\begin{array}{l}{a=2}\\{b=0}\end{array}\right.$；
综上所述：a、b的值为：$\left\{\begin{array}{l}{a=\frac{3}{2}}\\{b=\frac{1}{2}}\end{array}\right.$；$\left\{\begin{array}{l}{a=1}\\{b=1}\end{array}\right.$；$\left\{\begin{array}{l}{a=\frac{3}{2}}\\{b=-\frac{1}{2}}\end{array}\right.$；$\left\{\begin{array}{l}{a=1}\\{b=-1}\end{array}\right.$；$\left\{\begin{array}{l}{a=2}\\{b=0}\end{array}\right.$．

点评 本题主要考查了一元二次方程的概念．解题的关键是分5种情况讨论x的指数．