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Let \(a,b,c\) be the real roots of \(x^3 -4x^2 -32x+17=0.\) Solve for \(x\) in \(\sqrt[3]{x - a} + \sqrt[3]{x - b} + \sqrt[3]{x - c} = 0.\)
Please help!

 Jun 20, 2022
 #1
avatar+13799 
+2

Let a, b, c   be the real roots of   \(x^3 -4x^2 -32x+17=0.\)

Solve for  x in:                            \(\sqrt[3]{x - a} + \sqrt[3]{x - b} + \sqrt[3]{x - c} = 0.\)

 

Hello Guest!

 

WolframAlpha calculated:

a = - 4.3195

b = 0.50355

c = 7.8159

a + b + c = 4

abc = - 17

\(\sqrt[3]{x - a} + \sqrt[3]{x - b} + \sqrt[3]{x - c} = 0\\ \sqrt[3]{x +4.3195} + \sqrt[3]{x - 0.50355} + \sqrt[3]{x - 7.8159} = 0\)

No solution found.

laugh  !

 Jun 20, 2022
 #2
avatar+117752 
+2

Hi asinus,

That was good thinking.

 

But I graphed your equation using Desmos and there was a solution

 

 Jun 21, 2022
 #3
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+2

Hello Guest!

 

If a,b,c are the solutions of the polynomial equation:    \(x^3-4x^2-32x+17=(x-a)(x-b)(x-c)=0\) 

Then, by Vietas' formulae:                   \(a+b+c=4\)

                                                           \(ab+ac+bc=-32\) 

                                                                  \(abc=17\)

Using the following identity:  

\(y_1^3+y_2^3+y_3^3-3y_1y_2y_3=(y_1+y_2+y_3)(y_1^2+y_2^2+y_3^2-y_1y_2-y_1y_3-y_2y_3)\)  (*)

(More commonly written as: \(x^3+y^3+z^3=3xyz\) if and only if \(x+y+z=0\)).

 

Then:

Let  \(y_1=\sqrt[3]{x-a}\) , \(y_2=\sqrt[3]{x-b}\) , \(y_3=\sqrt[3]{x-c}\)

We are given: \(y_1+y_2+y_3=0\)

Hence, (*) becomes: 

          \((x-a)+(x-b)+(x-c)=3\sqrt[3]{(x-a)(x-b)(x-c)}\)

     

   (Notice: \((x-a)(x-b)(x-c)=x^3-4x^2-32x+17\))

 

Thus,

 

            \( 3x-(a+b+c) = 3\sqrt[3]{x^3-4x^2-32x+17}\)   

            (By Vietas formula: a+b+c=4, substitute and cube both sides)

 

\(\iff (3x-4)^3=27(x^3-4x^2-32x+17)\)

 

Expanding:

 

\(\iff 27x^3-108x^2+144x-64=27x^3-108x^2-864x+459\)

 

Simplify to get a linear equation:

 

\(144x-64=-864x+459 \implies 1008x=523 \implies x=\frac{523}{1008}\)

 

Therefore,  \(x=\dfrac{523}{1008}\)

 Jun 21, 2022
 #4
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0

Hi asinus, by the way, Wolframalpha approximated a,b,c a lot.

Because when using mathway.com, it gives the following values: 

\(a=-4.31946756\)

\(b=0.5035452\)

\(c=7.81592235\)

So:

\(\sqrt[3]{(x+4.31946756)}+\sqrt[3]{(x-0.5035452)}+\sqrt[3]{(x-7.81592235)}=0\)

And, then graphing this equation as Melody did (But using the more accurate values of a,b,c):

Giving the approximation: \(0.519\)   (And the exact solution is: \(\frac{523}{1008}=0.518849206..\)).

Hope this helps!

 Jun 21, 2022

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