In the following, determine whether the given quadratic equations have real roots and if so, find the roots: `2x^2+5sqrt(3)x+6=0` (ii) `sqrt(2)x^2+7x+5sqrt(2)=0`

Explanation

The given equation is `2x^(2)+5sqrt(3)x+6=0.`
Comparing it with `ax^(2)+bx+c=0,` we get
`a=2,b=5sqrt(3)" and "c=6.`
`:." "D=(b^(2)-4ac)=[(5sqrt(3))^(2)-4xx2xx6]=(75-48)=27gt0.`
So, the given equation has real roots.
Now, `sqrt(D)=sqrt(27)=3sqrt(3).`
`:." "alpha=(-b+sqrt(D))/(2a)=((-5sqrt(3)+3sqrt(3)))/(2xx2)=(-2sqrt(3))/(4)=(-sqrt(3))/(2),`
`beta=(-b-sqrt(D))/(2a)=((-5sqrt(3)-3sqrt(3)))/(2xx2)=(-8sqrt(3))/(4)=-2sqrt(3).`
Hence, `(-sqrt(3))/(2)` and `-2sqrt(3)` are the roots of the given equation.