Hello!
I need help with these two problems- they're driving me crazy. Any input would be really appreciated!!
This is the first question:
This is the second question:
Thanks so so much!
1) $\displaystyle x + \frac{1}{x} = 3$
$\displaystyle x^2 + 1 = 3x$
$\displaystyle x^2 - 3x + 1 = 0$
$\displaystyle x = \frac{3 \pm \sqrt{(-3)^2 - 4(1)(1)}}{2(1)}$
$\displaystyle = \frac{3 \pm \sqrt{5}}{2}$.
So $\displaystyle x = \frac{3 - \sqrt{5}}{2}$ or $\displaystyle x = \frac{3 + \sqrt{5}}{2}$.
2) $\displaystyle x^2 - 3x - 5 = 2$
$\displaystyle x^2 - 3x - 7 = 0$
$\displaystyle x = \frac{3 \pm \sqrt{(-3)^2 - 4(1)(-7)}}{2(1)}$
$\displaystyle = \frac{3 \pm \sqrt{37}}{2}$
So $\displaystyle x = \frac{3 - \sqrt{37}}{2}$ or $\displaystyle x = \frac{3 + \sqrt{37}}{2}$.
From the sum of cubes formula: $\displaystyle x^6 + \frac{1}{x^6} = (x^2)^3 + \left(\frac{1}{x^2}\right)^3 = \left( x^2 + \frac{1}{x^2}\right) \left( x^4 - 1 + \frac{1}{x^4}\right)$.
Now note:
1. $\displaystyle \left(x + \frac{1}{x} \right)^2 = x^2 + 2 + \frac{1}{x^2} = 9$. Therefore $\displaystyle x^2 + \frac{1}{x^2} = .... $
2. $\displaystyle \left(x^2 + \frac{1}{x^2} \right)^2 = x^4 + 2 + \frac{1}{x^4} = .... $ (use the above value). Therefore $\displaystyle x^4 + \frac{1}{x^4} = .... $
Dear carla5980,
Here's another approch,
$\displaystyle x+\frac{1}{x}=3$
$\displaystyle (x+\frac{1}{x})^3=27$
$\displaystyle x^3+\frac{1}{x^3}+3(x+\frac{1}{x})=27$
$\displaystyle x^3+\frac{1}{x^3}+(3\times{3})=27$
$\displaystyle x^3+\frac{1}{x^3}=27-9=18$
$\displaystyle (x^3+\frac{1}{x^3})^2=18^2$
$\displaystyle x^6+\frac{1}{x^6}+2=18^2$
$\displaystyle x^6+\frac{1}{x^6}=324-2=322$
Hope this helps.
Questions of this sort assume that you know and can do certain things. eg. Do you know the sum of cubes formula?
From 1. it should be obvious that $\displaystyle x^2 + \frac{1}{x^2} = 7$.
Using that result in 2, you get $\displaystyle \left(x^2 + \frac{1}{x^2} \right)^2 = x^4 + 2 + \frac{1}{x^4} = 49$ and so $\displaystyle x^4 + \frac{1}{x^4} = 47$.
Now you substitute 7 and 49 into $\displaystyle \left( x^2 + \frac{1}{x^2}\right) \left( x^4 - 1 + \frac{1}{x^4}\right)$ to get the answer: (7)(46) = 322.
Sorry, but you don't seem to have made much effort in trying to follow the help you were given and filling in the details.