Solve- 3(x^2+1/x^2)-16(x-1/x)+26=0 Ans.-1,3,1/3 Please tell me how to solve it.

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gsenviro | College Teacher | (Level 1) Educator Emeritus

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For the given question: `3(x^2 +1/x^2) -16(x-1/x) + 26 = 0`

The answers can not be -1, 3 or 1/3.

Pls. try substituting any of them and you will see that they do not satisfy the equation.

Let us assume `x-1/x = y`

Squaring both the sides, we get: `(x-1/x)^2 = y^2`

or, `x^2 + 1/x^2 -2 = y^2`

or, `y^2 + 2 = x^2 + 1/x^2`

substituting these values in the equation,we get

`3(y^2 + 2) -16y + 26 = 0`

or `3y^2 - 16y +32 = 0`

solving this quadratic equation, we get: `y = (16 +- sqrt(16^2 - 4*3*32))/(2*3)`

or, y =  `(16 +- 8sqrt 2 i)/6 = (8+-4sqrt2 i)/3`

now, y = x-1/x = `(8+-sqrt(2)i)/3`

This will be another quadratic equation set.

for example, let us take the first root of y, `x-1/x`  = `(8+sqrt(2)i)/3`  

on simplification we get: 3(x^1-1) = x(8+`sqrt2` i) 

i.e., 3x^2 -(8+i`sqrt2` )x -3 = 0

we can solve this to get some values of x and similarly try the other root of y for more values of x.

As you can see that -1,3 and 1/3 are not the solutions. Pls. check the equation again.

Try and solve these quadratic equation to get the answer.

Hope this helps.

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tonys538 | Student, Undergraduate | (Level 1) Valedictorian

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The solution you have provided are not of the equation 3(x^2+1/x^2)-16(x-1/x)+26=0. Rather they are the solutions of the equation 3(x^2+1/x^2)-16(x+1/x)+26=0. You have made a typo with the minus sign.

The equation 3(x^2+1/x^2)-16(x+1/x)+26=0 can be solved as follows.

Let x + 1/x = y, squaring both the sides x^2 + 1/x^2 + 2 = y^2 or x^2 + 1/x^2 = y^2 - 2

Substituting y in the original equation:

3(y^2 - 2) - 16y+26 = 0

3y^2 - 6 - 16y + 26 = 0

3y^2 - 16y + 20 = 0

3y^2 - 6y - 10y + 20 = 0

3y(y - 2) - 10(y - 2) = 0

(3y - 10)(y - 2) = 0

y = 10/3, y = 2

y = x + 1/x

x + 1/x = 10/3

3x^2 - 10x + 3 = 0

3x^2 - 9x - x + 3 = 0

3x(x - 3)-1(x - 3) = 0

(3x - 1)(x - 3) = 0

x = 1/3, x = 3

x + 1/x = 2

x^2 - 2x + 1 = 0

(x - 1)^2 = 0

x = 1

The solution of the equation 3(x^2+1/x^2)-16(x+1/x)+26=0 are {1, 1/3, 3}

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