Question

x= 1/2-root 3 find the value of x^3 -2x^2 - 7x +5

Answer 1

What is the sum of the square of the roots of the equation x^2-7 [x] +5.?

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1 ANSWER



Pranav Nair

Answered Oct 3 2015

[Note: [x] is the integral part of x. For eg. if x=2.3 then[x]=2, if x=-2.3 then [x]=-3, and if x=3 then [x]=3 ]

Let us break this equation into two.

y=x^2+5 ----- (1)   and 
y=7[x] -----(2)
minimum value of (1) is 5 at x=0 and (2) is less than zero for negative values of x concludes that roots are going to lie entirely in the first quadrant.

Let us start inspection piece-wise.

0<x<1  (2) is zero and (1) is greater than 5 
=> NO SOLUTION

1<x<2 : [x]=1 and hence 7[x]=7.
Putting x^2+5=7 we get x^2=2
Since x=1.414 and [x]=1 we have one solution in this interval. (x1)
=>x1^2=2

2<x<3: [x]=2 and hence 7[x]=14
Putting x^2+5=14 we get x=3. But then [x] is 3 not 2.
=>NO SOLUTION

3<x<4: [x]=3 and hence 7[x]=21
Putting x^2+5=16 we get x=4. But then [x] is 4 not 3.
=>NO SOLUTION

4<x<5: [x]=4 and hence 7[x]=28
Putting x^2+5=28 we get x^2=23. 
Since x=4.796 and [x]=4 we have one solution in this interval. (x2)
=> x2^2=23

5<x<6: [x]=5 and hence 7[x]=35
Putting x^2+5=35 we get x^2=30. 
Since x=5.477 and [x]=5 we have one solution in this interval. (x3)
=> x3^2=30

6<x<7: [x]
7<x<8: [x]=7 and hence 7[x]=49
Putting x^2+5=49 we get x^2=44. 
But here x=6.633 and [x]=6 and not 7.
=> NO SOLUTION

8<x<9: [x]=8 and hence 7[x]=56
Putting x^2+5=56 we get x^2=51. 
But here x=7.141 and [x]=7 and not 8.
=> NO SOLUTION

And likewise both curves diverges as we go further as x^2+5 grows quickly compared to 7[x]. 

Thus sum of squares of the equation x^2-7[x]+5=0 is
x1^2+x2^2+x3^2+x4^2
=> 2+23+30+37=92

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Answer 2

hope it help you. . . .