a*****g
发帖数: 1320 | 1 【 以下文字转载自 SanFrancisco 讨论区 】
发信人: ashuang (东洛杉矶推妈+股票大妈), 信区: SanFrancisco
标 题: mathscounts 数学题求助(4)
发信站: BBS 未名空间站 (Wed Jan 19 00:02:32 2011, 美东)
1) The solutions of x2 +bx+c=0 are each 5 more than the solution of x2+7x+3=0. What are the value of b and c? Express your answer as an ordered pair (b,
c)
2) A cubic equation of the form x3+ bx2+cx+d==0 has solutions x=3, x=4 and x
=5. What are the value of b, c, and d? Express your answer as an ordered
triple (b,c,d)
3) What is the sum of the reciprocals of the solutions of x3-3x2-13x+15=0?
Express your answer as a common fraction
4) What is the sum of the squares of the solutions of x3-15x2+66x-80=0?
5) The solutions of X3-63x2+cx-1728=0 form a geometric sequence. What is the
value of c?
Thanks a lot! | y****i
发帖数: 12114 | 2 1. （-3，-7）
2. (-12,47,-60) | B******1
发帖数: 9094 | 3 Without my glasses on, I read "What is the sum" as "What is the scum!" | B******1
发帖数: 9094 | 4 3) What is the sum of the reciprocals of the solutions of x3-3x2-13x+15=0?
Express your answer as a common fraction
Apparently X=1 is a solution. Therefore,
x3-3x2-13x+15=(x-1)(X2-2x-15)=(x-1)(X-5)(X+3)
1/1 + 1/5 +(-1/3)= 13/15
Another way to solve this is to realize that if x1, x2, and x3 are the three
roots of this equation,
then, x1*x2*x3 = -15, and x1*x2+x1*x3 +x2*x3 = -13.
Hence, 1/x1 + 1/x2 + 1/x3 = (-13)/(-15)=13/15 | B******1
发帖数: 9094 | 5 What is the sum of the squares of the solutions of x3-15x2+66x-80=0?
(x1)^2 + (x2)^2 + (x3)^2 = (x1 + x2 + x3)^2 - 2*(x1x2 + x1x3 + x2x3)
= 80^2 -2*(66)=6400 - 132 = 6268 | B******1
发帖数: 9094 | 6 The solutions of X3-63x2+cx-1728=0 form a geometric sequence. What is the
value of c?
roots: x, x^2, x^3
x*x^2*x^3=1728; x*x^2 + x*x^3 + x^2*x^3 = C;
x^2=12
C=12(2*3^(1/2) +12 + 24*3^(1/2))=144 + 312*3^(1/2) | Y****a
发帖数: 243 | 7 geometric sequence 不是 x x^2 & x^3 吧
应该是 x, xy, and xyy,
我算的 这个方程的解是 3，12，和48
【在 B******1 的大作中提到】
: The solutions of X3-63x2+cx-1728=0 form a geometric sequence. What is the
: value of c?
: roots: x, x^2, x^3
: x*x^2*x^3=1728; x*x^2 + x*x^3 + x^2*x^3 = C;
: x^2=12
: C=12(2*3^(1/2) +12 + 24*3^(1/2))=144 + 312*3^(1/2)
| u*h
发帖数: 397 | 8 本着授人以渔的方针， 解释一下，
对于一个三元一次方程（其他高次方程类似）：
X^3 + aX^2+bX+c = 0
如果它的解是:x1, x2, and x3;
那么有如下关系：
(X-x1)*(X-x2)*(X-x3) = X^3 + aX^2+bX+c
将方程左边展开， 合并同类项， 对照等号左右两边，
可以得出三个关系式。
对照题目的要求， 通过这三个关系式的一些简单组合，
一般就可以得到题目所求的答案。
本类题目的解法均类似。
点到这里， 小孩就应该能够自己完成这些题目了。 | a*****g
发帖数: 1320 | 9 Super many thanks you guys great help!! |
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