Show that the equation `2x^(2)-6x+3=0` has real roots and find these roots.

Explanation

The given equation is `2x^(2)-6x+3=0.`
This is of the form `ax^(2)+bx+c=0,` where a=2, b=-6 and c=3.
`:." "D=(b^(2)-4ac)={(-6)^(2)-4xx2xx3}=(36-24)=12gt0.`
So, the given equation has real unequal roots.
Solving `2x^(2)-6x+3=0` by quadratic formula, we have
`x=(6+-sqrt(36-4xx2xx3))/((2xx2))=(6+-sqrt(36-24))/(4)=(6+-sqrt(12))/(4)`
`implies" "x=(6+-2sqrt(3))/(4)impliesx=(3+-sqrt(3))/(2).`
So, `((3+sqrt(3)))/(2)" and "((3-sqrt(3)))/(2)` are the roots of the given equation.