Which of the following equations has no real roots?

Options

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

Answer

A

Explanation

(a) The given equation is `x^(2)-4x+3sqrt(2)=0`
On comparing with `ax^(2)+bx+c=0`, we get
`a=1,b=-4` and `c=3sqrt(2)`
The discriminant of `x^(2)-4x+3sqrt(2)=0` is
`D=b^(2)-4ac`
`=(-4)^(2)-4(1)(3sqrt(2))=16-sqrt(2)=16-12xx(1.41)`
`=16-16.92=-0.92`
`implies b^(2)-4aclt0`
(b) The given equation is `x^(2)+4x-3sqrt(2)=0`
On comparing the equation with `ax^(2)+bx+c=0` we get
`a=1,b=4` and `c=-3sqrt(2)`
Then `D=b^(2)-4ac=(-4)^(2)-4(1)(-3sqrt(2))`
`=16+12sqrt(2)gt0`
Hence the equation has real roots.
(c) Given equation is `x^(2)-4x-3sqrt(2)=0`
On comparing the equation with `ax^(2)+bx+c=0` we get
`a=1,b=-4` and `c=-3sqrt(2)`
Then `D=b^(2)-4ac=(-4)^(2)-4(1)(-2sqrt(2))`
`=16+12sqrt(2)gt0`
Hence, the equation has real roots.
(d) Given equation is `3x^(2)+4sqrt(3)x+4=0`
On comparing the equation with `ax^(2)+bx+c=0` we get
`a=3,b=4sqrt(3)` and `c=4`
Then `D=b^(2)-4ac=(4sqrt(3))^(2)-4(3)(4)=48-48=0`
Hence, the equation has real roots.
Hence `x^(2)-4x+3sqrt(2)=0` has no real roots.