Question

$\text{If}{x}^3+\frac{1}{x^3}=110,\text{then}x+\frac{1}{x}=$ ?

• A. 5
• B. 10
• C. 15
• D. none of these

Answer 1

Answer: (A)

$x^3+\frac{1}{x^3}=110$

Use the algebraic identity: $a^3+b^3=(a+b{)}^3-3ab(a+b)$

$\Rightarrow x^3+\frac{1}{x^3}={(x+\frac{1}{x})}^3-3.x.\frac{1}{x}(x+\frac{1}{x})\Rightarrow 110={(x+\frac{1}{x})}^3-3(x+\frac{1}{x})Letx+\frac{1}{x}=a,\text{then}{a}^3-3a-110=0\text{Factors of}-110=\pm 1,\pm 2,\pm 5,\pm 11\text{Let}a=1,\text{then}1-3-110\ne 0\text{Let}a=-1,\text{then}-1+3-110\ne 0Leta=2,\text{then}8-6-110\ne 0\text{Let}a=-2,\text{then}-8+6-110\ne 0Leta=5,\text{then}125-15-110=0$

$\therefore a=5$

$\Rightarrow x+\frac{1}{x}=5$