Find the nature of the roots of the following quadratic equations. If the real roots exist, find them:(i) `2x^2-3x+5=0`(ii) `3x^2-4sqrt(3)x+4=0`(iii) `2x^2-6x+3=0`

Explanation

(i) Given equation is `2x^(2)-3x+5=0`
Comparing with `ax^(2)+bx+c=0`, we get
a=2, b=-3 and c=5
`:."Discriminant"D=b^(2)-4ac`
`impliesD=(-3)^(2)-4xx2xx5`
`impliesD=9-40=-31lt0`
Hence, the equation has no real roots i.e., it has imaginary roots.
(ii) Given equation is
`3x^(2)-4sqrt3x+4=0`
Comparing with `ax^(2)+bx+c=0`, we get
`a=3,b=-4sqrt3andc=4`
`:."Discirminant"D=b^(2)-4ac`
`impliesD=(-4sqrt3)^(2)-4xx3xx4`
`impliesD=48-48=0`
Hence, the equation has two real and equal roots.
When D=0 roots of the equation are `((-b))/(2a)`
`:."Roots are"(-(-4sqrt3))/(2xx3)=(4sqrt3)/(6)=(2sqrt3)/(3)`
The two equal roots `=(2sqrt3)/(3)and(2sqrt3)/(3)`
(iii) Given equation is `2x^(2)-6x+3=0`
On comparing with `ax^(2)+bx+c=0`, we get
a=2,b=-6, and c=3
`:."Discriminant"D=b^(2)-4ac`
`=(-6)^(2)-4xx2xx3=36-24=12gt0`
Hence, the equation has two distinct real roots
`:.x=(-b+-sqrtD)/(2a)=(-(-6)+-sqrt12)/(2xx2)=(6+-2sqrt3)/(4)=(3+-sqrt3)/(2)`
Hence, the real roots are `(3+sqrt3)/(2)and(3-sqrt3)/(2)`.