问题 解答题

已知关于x的方程x2+x+a-a2=0和x2-(3a-1)x+(2a+1)(a-2)=0.问是否存在这样的a值,使得第一个方程的两实根的平方和等于第二个方程的一个整数根?若存在,求出这样的a值;若不存在,请说明理由.

答案

第一个方程x2+x+a-a2=0,即有(x+a)(x+1-a)=0,

∴x1=-a,x2=a-1,

故x12+x22=a2+(a-1)2=2a2-2a+1,

由第二方程x2-(3a-1)x+(2a+1)(a-2)=0,

得[x-(2a+1)][x-(a-2)]=0,

x3=2a+1,x4=a-2,

若x3为整数,则2a2-2a+1=2a+1,解得a=0或2,此时x3=1或5,

若x4为整数,则2a2-2a+1=a-2,即2a2-3a-3=0,此方程无实数根,

综上可知,当a=0或2时,第一个方程的两个实数根的平方和等于第二个方程的一个整数根.

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Student Services

Students' Union

Every University has a Students' Union. If students are the lifeblood of a university, the Union is surely its beating heart.

For students, by students

A Students’ Union is run by students and acts independently. At the end of each academic year, it will find six students to hold a sabbatical(带有休假的) position managing the Union for the coming year. That the six successful sabbatical officers have only recently experienced issues affecting students ensures that your needs are very well understood.

Letting your hair down

Although the University is spread across a fairly wide area, the Union is centrally located and is only a short walk from any of the main buildings. This makes it very easy to catch up with friends who might be on a different course.

At night, the place really comes alive – we have three different bars to suit every occasion. The venues are student-only so you can be sure of meeting people with a similar outlook and we throw in a transport service to make sure you get home safely.

The Union also organizes various formal balls throughout the year, with the highlight being the end-of-year ball.

Supporting you

Beyond having fun, we also make sure some of the more serious issues are effectively dealt with. If you have a problem with your course, another sabbatical officer is trained to help you. Being independent from the University means we can represent and support you to the best of our ability, and once you reach university you may find that you become aware of more national or political issues.

小题1:From the second paragraph of the passage, we can infer that ______.

A.a Students’ Union is operated by the university

B.a Students’ Union is not operated by the university

C.a Students’ Union is operated is working for the university

D.a Students’ Union is affected by the university小题2: The person who takes up a certain post of the Students’ Union ______.

A.learns what the students need

B.is very happy and easy

C.can affect the life of any student

D.has understood everything of the University小题3:According to the third paragraph, we can learn that ______.

A.a Students’ Union is far away from the university

B.a Students’ Union is near to the home

C.a Students’ Union is central part of the university

D.a Students’ Union has different courses for all the students小题4:From the passage, we can learn that ______ is the most wonderful program of every year.

A.making friends

B.dancing in a bar

C.the match of the end-year ball

D.electing the officers小题5:After reading the last two parts, we can infer that _______.

A.the Students’ Union can help a student who has trouble in the study

B.the Students’ Union can help a student who is being trained

C.the Students’ Union can cultivate the ability of a student to be suitable for political position

D.the Students’ Union can support a student to run the University