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Tackling Algebra

I AM UNABLE TO TACKLE THIS PROBLEM. CAN ANYONE COME FORWARD TO HELP ME.

a+b+c=-1/2,ab+bc+ca=-1/2,abc=1/8, then how to get 1) 1/a+1/b+1/c and 2)a^1/3+b^1/3+c^1/3.

Note by Prabir Chaudhuri
2 years, 8 months ago

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Let \[a^{\frac{1}{3}} = x, b^{\frac{1}{3}} = y ,c^{\frac{1}{3}} = z\]

then we have

\[\begin{cases} x^3+y^3+z^3=-\dfrac{1}{2}\\ (xy)^3+(yz)^3+(xz)^3=-\dfrac{1}{2}\\ (xyz)^3=\dfrac{1}{8} \end{cases}\]

using the identity

\[A^3+B^3+C^3=(A+B+C)^3-3(A+B+C)(AB+BC+AC)+3ABC\]

we have,

\[\begin{cases} (x+y+z)^3-3(x+y+z)(xy+yz+xz)+3xyz=-\dfrac{1}{2}\\ (xy+yz+xz)^3-3(xy+yz+xz)[xyz(x+y+z)]+3x^2y^2z^2=-\dfrac{1}{2}\\ xyz=\dfrac{1}{2} \end{cases}\]

Now,

Let \[u=x+y+z, v=xy+yz+xz\]

then we have

\[\begin{cases} u^3-3uv+2=0\\ 4v^3-6uv+5=0 \end{cases}\]

\[\Longrightarrow 4v^3-2u^3+1=0, v=\dfrac{u^3+2}{3u}\]

so

\[4\left(\dfrac{u^3+2}{3u}\right)^3-2u^3+1=0\Longrightarrow 4u^9-30u^6+75u^3+32=0\]

let \(t=u^3\), so we have

\[4t^3-30t^2+75t+32=0\]

let \(t=\dfrac{5}{2}-a\),then

\[4\left(\dfrac{5}{2}-a\right)^3-30\left(\dfrac{5}{2}-a\right)^2+75\left(\dfrac{5}{2}-a\right)+32=0\]

\[\Longrightarrow 4a^3=\dfrac{189}{2}\Longrightarrow a=\dfrac{3\sqrt[3]{7}}{2}\]

\[\Longrightarrow t=\dfrac{5}{2}-\dfrac{3\sqrt[3]{7}}{2}\]

Therefore,

\[a^{\frac{1}{3}} + b^{\frac{1}{3}} + c^{\frac{1}{3}} = x+y+z = u = t^{\frac{1}{3}} = \sqrt[3]{\dfrac{1}{2}\left(5-3\sqrt[3]{7}\right)} \] Ishan Singh · 2 years, 7 months ago

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@Ishan Singh @PRABIR CHAUDHURI I have posted my solution of the other part. You may look at it. Ishan Singh · 2 years, 7 months ago

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See.. the first part is easy as ab+bc+ca=-1/2divide the equation by abc which is also equal to 1/8 . We get 1/a +1/b +1/c=1/2 * 8 =4. Mayur Ade · 2 years, 8 months ago

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@Mayur Ade Thanks. Prabir Chaudhuri · 2 years, 8 months ago

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Comment deleted Jan 09, 2015

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@U Z I think its \( a^{\frac{1}{3}} \) , not \( a^{-3} \). Siddhartha Srivastava · 2 years, 8 months ago

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@Siddhartha Srivastava Oh yes sorry its totally wrong , it was 12:30 nearly I was was half sleep , and I wrote it sorry U Z · 2 years, 8 months ago

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@Siddhartha Srivastava Please note that it is a^(1/3),not a^(-3) as anticipated. Is there any general formulation for evaluating a^(1/n) +b^(1/n) + c^(1/n) with the input as given with the problem ?Thanks. Prabir Chaudhuri · 2 years, 8 months ago

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@Siddhartha Srivastava Yeah, you are right. It is a^(1/3),not a^(-3). Prabir Chaudhuri · 2 years, 8 months ago

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1/a+1/b+1/c=(ab+bc+ca)/abc Sanchayan Dutta · 1 year, 11 months ago

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