Problem Solving (PS) | Re: What is the equation whose roots are and , if (-^2/) and (-^2/

What is the equation whose roots are α and β, if\((\frac{-α^2}{β})\) and\((\frac{-β^2}{α})\) are roots of the equation\( 3x^2 – 18x + 2 =0\) ?


A.\( 3x^2 + 6x + 2 =0\)

B.\( 3x^2 - 6x + 2 =0\)

C.\( 3x^2 + 6x - 2 =0\)

D.\( 3x^2 - 6x - 2 =0\)

E.\( 3x^2 - 6x - 3 =0\)

­I believe there is an error somewhere. The math ain't mathing. I could be misguided in my math here.

For simplicity, I will refer alpha as 'a' and beta as 'b'. 

From equation 2:

a)Product:
\(\frac{2}{3} =(\frac{-a^2}{b})(\frac{-b^2}{a})\)
\(\frac{2}{3} =-a*-b\)
\( 2 =3ab\)
\( 4 =6ab\) ----a

b)Sum:
\(\frac{18}{3} =(\frac{-a^2}{b} ) +(\frac{-b^2}{a})\)
\( 6 =\frac{ (-a^3 -b^3)}{ab}\)
\( 6ab = -a^3 -b^3\) (substitute the value of6ab)
\( 4 = -a^3-b^3\) ----b

TheRationale: 

From eqa , for 6ab to result in positive number, either both a & b must be negative or bothpositive . Now eqb helps us narrow downfurther:  either both a^3 and b^3 must benegative to give a positive sum, OR the larger one of them should benegative .

So using eq a and b together, we get to know that both a (alpha) and b (beta) must be negative. 

However,
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Statistics : Posted by Bunuel • on 13 Jul 2023, 01:30 • Replies 3 • Views 1412

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