Question

Let u and v be the solutions to 3x^2 + 5x + 7 = 0. Find u/v+v/u

Answer 1

Answer:
`(-17)/(21)`

Explanation:

Given: u and v be are the solutions of
`3x^2+5x+7=0`

Let
`ax^2+bx+c=0` is the quadratic equation and u and v are the zeroes/solutions then

Sum of zeroes;
`u+v = (-b)/(a)`

Product of zeroes;
`uv= (c)/(a)`

Comparing
`3x^2+5x+7=0` to
`ax^2+bx+c=0`

we get a= 3 , b= 5 and c = 7

`u+v = (-b)/(a) = (-5)/(3)----(i)`

`uv= (c)/(a) = (7)/(3)----(ii)`

Now we have to find

`(u)/(v) +(v)/(u) =(u^2+v^2)/(uv)` adding and subtracting 2uv in numerator we get

`= (u^2+v^2+2uv-2uv)/(uv)= ((u+v)^2-2uv)/(uv)`

Substituting the values from (i) and (ii) we get

`(((-5)/(3) )^2-2* (7)/(3) )/((7)/(3) ) = ((25)/(9) -(14)/(3) )/((7)/(3) )= ((25-42)/(9) )/((7)/(3)) =(-17)/(9) * (3)/(7) = (-17)/(21)`

Hence, the value of
`(u)/(v) +(v)/(u)` is
`(-17)/(21)`