question

Asked by arindeep.singh | 26th Sep, 2020, 10:39: AM

Expert Answer:

Q: Does there exist a quadratic equation whose coefficients are all distinct irrationals but both the roots are rationals? Why?
Solution:
Yes, it is possible that a quadratic equation with distinct irrational coefficients have rational roots.
It is because, the roots are rational if the discriminant is a perfect square. So, it depends upon the discriminant and not on the coefficients.
 
Here is an example:
square root of 3 straight x squared minus square root of 27 straight x plus square root of 12 equals 0
straight x equals fraction numerator negative straight b plus-or-minus square root of straight b squared minus 4 ac end root over denominator 2 straight a end fraction
equals fraction numerator square root of 27 plus-or-minus square root of 27 minus 4 open parentheses square root of 3 close parentheses open parentheses square root of 12 close parentheses end root over denominator 2 square root of 3 end fraction
equals fraction numerator 3 square root of 3 plus-or-minus square root of 27 minus 24 end root over denominator 2 square root of 3 end fraction
equals fraction numerator square root of 3 open parentheses 3 plus-or-minus 1 close parentheses over denominator 2 square root of 3 end fraction
equals fraction numerator 3 plus-or-minus 1 over denominator 2 end fraction
equals 2 space or space 1

Answered by Renu Varma | 28th Sep, 2020, 11:00: AM