Question

If x + 1/x = square root of 5, find
x^4 + x^4 +1/x^4

​

Answer 1

Answer:

x+1x=5

then

(x+1x)2=25

or

x2+2+1x2=25

or

x2+1x2=23

therefore

(x2+1x2)2=232

or

x4+2+1x4=529

or

x4+1x4=527.

Let’s try the direct method.

x+1x=5

therefore

x2–5x+1=0.

(One should say that x≠0 , but x=0 does not satisfy the equation.)

The (two) possible values for x are

x=12(5±21−−√).

Let’s calculate x4 and 1/x4 .

First notice that the product of both roots are 1, then one is the reciprocal of the other, and the roles of x4 and 1/x4 are interchanged.

(5+21−−√2)(5−21−−√2)

=25–214=1.

(We could had seen that from the independent term in the equation above, which is the product of the roots).

Now let’s calculate the fourth power of the roots.

First the square

(5±21−−√2)2

=25±1021−−√+214

=23±521−−√2.

Now the fourth power

(5±21−−√2)4

=(23±521−−√2)2

=232±23021−−√+25⋅214

=527±11521−−√2.

Last step, add the fourth power of the roots together

527+11521−−√2+527−11521−−√2=527.

Step-by-step explanation:

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