The roots of the quadratic equation x^2-5x+6=0 are α and β.The value of 1/(α^2) +1/(β^2) is13/36.Show that α^4-65α+114=0.Thanks.

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The roots of the quadratic equation x^2 - 5x + 6 = 0 are `alpha` and `beta` . The value of `1/(alpha^2) +1/(beta^2) = 13/36` .

`x^2 - 5x + 6 = (x - alpha)(x - beta)`

=> `x^2 - 3x - 2x + 6 = (x - alpha)(x - beta)`

=> `x(x - 3) - 2(x - 3) = (x - alpha)(x - beta)`

=> `(x - 3)(x - 2) = (x - alpha)(x - beta)`

`alpha = 3` and `beta = 2`

These values of `alpha` and `beta` give: `1/alpha^2 + 1/beta^2 = 1/4 + 1/9 = 13/36`

`alpha^4 - 65*alpha + 114`

= `3^4 - 65*3 + 114`

= 81 - 195 + 114

= 0

This shows the value of the expression `alpha^4 - 65*alpha + 114 = 0`

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steamgirl | Student, College Junior | (Level 1) Honors

Posted on

Well first, by factoring x^2 -5x +6, you get the factors (x-3)(x-2), or x =3,2. Now plugging these into the equation 1/(α^2) +1/(β^2) = 13/36, works, but it won't matter which you chose for a or b, so plugging them into the final equation of α^4-65α+114=0, is the only way to determine which of the roots it is. In this case plugging in 2 will equal to zero, therefore a = 2 and b = 3.

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