If x^3 + 1/x^3 = 110 then find x + 1/x

Question

25-08-2016
Mathematics

contestada

If x^3 + 1/x^3 = 110 then find x + 1/x

Respuesta :

Otras preguntas

Your principal has decided to provided a treat to everyone in your grade before thanksgiving break-ice cream bears. one box contains 48 bars. There are 338 stud

The ability of children and adults to naturally try to deal with their problems is known as:

Khan academy algebra video why we do the same thing to both sides of an equation

Solving equations with variable on both sides Solve equation.check answer 5(y - 4)=7(2y+1) Pls & thank you again

HELP FAST Which of these provides the best evidence for the continental drift theory? variations in the types of fossils found in different continents variat

Invulnerability, where members of a group feel they can do no wrong, is a characteristic of

plants can often be propagated from stems but rarely from roots what features of shoots account for this difference?

WILL AWARD BRAINLIEST FOR YOUR HELP Catherine needs to sell boxes of cookies as a fundraiser for a team. She starts with 135 boxes and begins selling at a const

What was the name for the full-length nonreligious play of the spanish golden age?

Find the point, M, that divides segment AB into a ratio of 5:6 if A is at (0, 22) and B is at (11, 0). A) (5, 12) B) (5, 11) C) (6, 12) D) (6, 11)

dalendrk dalendrk · Answer 1 · 2016-08-25T19:24:18+02:00

[tex]Use:(a+b)^3=a^3+3a^2b+3ab^2+b^3\ (*)\\\\x^3+\left(\dfrac{1}{x}\right)^3\\=\underbrace{x^3+3\cdot x^2\cdot\dfrac{1}{x}+3\cdot x\cdot\left(\dfrac{1}{x}\right)^2+\left(\dfrac{1}{x}\right)^3}_{(*)}-3\cdot x^2\cdot\dfrac{1}{x}-3\cdot x\cdot\left(\dfrac{1}{x}\right)^2\\\\=\left(x+\dfrac{1}{x}\right)^3-3x-3\cdot\dfrac{1}{x}=\left(x+\dfrac{1}{x}\right)^3-3\left(x+\dfrac{1}{x}\right)[/tex]

therefore

[tex]x^3+\left(\dfrac{1}{x}\right)^3=110\iff \left(x+\dfrac{1}{x}\right)^3-3\left(x+\dfrac{1}{x}\right)=110\\\\subtitute\ t=x+\dfrac{1}{x}\\\\t^3-3t=110\ \ \ \ |subtract\ 110\ from\ both\ sides\\\\t^3-3t-110=0\\\\t^3-25t+22t-110=0\\\\t(t^2-25)+22(t-5)=0\ \ \ |use\ a^2-b^2=(a-b)(a+b)\\\\t(t-5)(t+5)+22(t-5)=0\\\\(t-5)[t(t+5)+22]=0\iff t-5=0\ or\ \underbrace{t^2+5t+22=0}_{no\ solution}\\\\t=5\iff x+\dfrac{1}{x}=5\\\\Answer:\boxed{x+\dfrac{1}{x}=5}[/tex]