Which of the following equations has two distinct real roots?

Options

A. `2x^(2)-3sqrt(2)x+9/4=0`
B. `x^(2)+x-5=0`
C. `x^(2)+3x+2sqrt(2)=0`
D. `5x^(2)-3x+1=0`

Answer

B

Explanation

The given equation is `x^(2)+x-5=0`
On comparing with `ax^(2)+bx+c=0` we get
`a=1, b=1` and `c=-5`
The discriminant of `x^(2)+x-5=0` is
`D=b^(2)-4ax=(1)^(2)-4(1)(-5)`
`=1+20=21`
`implies b^(2)-4acgt0`
So `x^(2)+x-5=0` has two distinct real roots.
(a) Given equation is `2x^(2)-3sqrt(2)x+9//4=0`
On comparing with `ax^(2)+bx+c=0`
`a=2,b=-3sqrt(2)` and `c=9//4`
Now `D=b^(2)-4ac=(-3sqrt(2))^(2)-4(2)(9//4)=18-18=0`
Thus, the equation has real and equal roots.
(c) Given equation is `x^(2)+3x+2sqrt(2)=0`
On comparing with `ax^(2)+bx+c=0`
`a=1,b=3` and `c=2sqrt(2)`
Now `D=b^(2)-4ac=(3)^(2)-4(1)(2sqrt(2))=9-8sqrt(2)lt0`
`:.` Roots of the equation are not real.
(d) Given equation is `5x^(2)+3x+1=0`
On comparing with `ax^(2)+bx+c=0`
`a=5,b=-3, c=1`
Now `D=b^(2)-4ac=(-3)^(2)-4(5)(1)=9-20lt0`
Hence roots of the equation are not real.