# If 2((x^2)+1)=5x, find i) x−(1/x) ii) x^3−(1/x^3)

If $2\left(\left({x}^{2}\right)+1\right)=5x$, find $i\right)x-\left(\frac{1}{x}\right)ii\right){x}^{3}-\left(\frac{1}{{x}^{3}}\right)$
You can still ask an expert for help

• Questions are typically answered in as fast as 30 minutes

Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it

Corben Pittman

We need to first find the value of xx by solving $2\left({x}^{2}+1\right)=5x.$
To solve $2\left({x}^{2}+1\right)=5x$, distribute the 2 to get $2{x}^{2}+2=5x.$
Subtracting 5x5x on both sides then gives $2{x}^{2}-5x+2=0.$
Factoring would then gives $\left(2x-1\right)\left(x-2\right)=0.$
Now that the equation is in factored form, we can set each factor equal to 0 to find the possible values of x:
2x−1=0 \ x−2=0
2x=1 \ x=2
$x=\frac{1}{2}$
Now that we have the possible values of x, we can substitute them into each expression and then simplify to find the values:
$i\right)Ifx=\frac{1}{2}$, then $x-\left(\frac{1}{x}\right)=\frac{1}{2}-\left(\frac{1}{1}/2\right)=\frac{1}{2}-2=-\left(\frac{3}{2}\right)$
If x=2, then $x-\left(\frac{1}{2}\right)=2-\left(\frac{1}{2}\right)=\frac{3}{2}$
ii)If $x=\frac{1}{2},then\left({x}^{3}\right)-\left(\frac{1}{{x}^{3}}\right)={\left(\frac{1}{2}\right)}^{3}-\left(\frac{1}{{\left(\frac{1}{2}\right)}^{3}}\right)=\frac{1}{8}-\left(\frac{1}{1}/8\right)=\left(\left(\frac{1}{8}\right)-8\right)=-\left(\frac{63}{8}\right)$
If x =2, then $\left({x}^{3}\right)-\left(\frac{1}{{x}^{3}}\right)=\left({2}^{3}\right)-\left(\frac{1}{{2}^{3}}\right)=8-\left(\frac{1}{8}\right)=\frac{63}{8}$