Question

If x³ + 1/x³ = 18 then x + 1/x = ?​

Answer 1

Step-by-step explanation:

What is x3+1/x3=18, then what is x+1/x?

I presume that, by ‘x3’, you mean ‘x^3’, i.e. ‘x cubed’.

As we’re interested in x+1/x not x , let’s use a substitution:

u=x+1x=x+x−1

Now, what happens when you cube u ?

From the binomial theorem, we have: (a+b)3=a3+3a2b+3ab2+b3

Using a=x and b=x−1 , we thus have:

u3=x3+3x+3x−1+x−3=x3+x−3+3(x+x−1)

=x3+x−3+3u⇒x3+x−3=u3−3u

From the question, we have; x3+x−3=18 , thus we have the cubic equation:

u3−3u−18=0

From inspection, it’s quite easy to spot that one solution is u=3 , thus (u−3) is a factor of the cubic expression. Rewriting the equation:

u3−3u−18=(u−3)(u2+3u+6)=0

For the other possible values of u , we just need to solve the quadratic equation u2+3u+6=0

Using the formula, we have:

u=−3±32−4×1×6√2×1=−3±9−24√2

=−3±−15√2=−3±15√i2

Thus the three possibilities for x+x−1 are:

3

−1.5+3.75−−−−√i

−1.5−3.75−−−−√i