Find the roots of the following quadratic equations, if they exist, by the method of completing the square:(i) `2x^2-7x+3=0` (ii) `2x^2+x-4=0`(iii) `4x^2+4sqrt(3)x+3=0` (iv) `2x^2+x+4=0`

Explanation

Given equation is
`2x^(2)-7x+3=0`
Dividing both sides by 2
`x^(2)-(7)/(2)x+(3)/(2)=0`
Adding `(("coefficient of x")/(2))^(2)" ""on both sides,i.e.,"((7)/(2)xx(1)/(2))^(2)=(49)/(16)`
`x^(2)-(7)/(2)x+(49)/(16)=(-3)/(2)+(49)/(16)`
`implies(x-(7)/(4))^(2)=((5)/(4))^(2)`
`:.x-(7)/(4)+-(5)/(4)" "(why?)`
`impliesx=(7)/(4)+-(5)/(4)=(7+-5)/(4)`
`impliesx=(12)/(4),(2)/(4)`
`impliesx=3,(1)/(2)`
Hence, roots of the equation are `(1)/(2)` and 3.
(ii) Given equation is `4x^(2)+4sqrt3x+3=0`
Dividing both sides by 4
`impliesx^(2)+sqrt3x+(3)/(4)=0`
`impliesx^(2)+sqrt3x=(3)/(4)`
Adding `(("coefficient of x")/(2))^(2)i.e.,((sqrt3)/(2))=(3)/(4)"on both sides"`
`x^(2)+sqrt3x+(3)/(4)=(-3)/(4)+(3)/(4)`
`implies(x+(sqrt3)/(2))^(2)=0`
`:.(x+(sqrt3)/(2))=0and((x+(sqrt3)/(2)=0`
`impliesx=(-sqrt3)/(2)andx=(-sqrt3)/(2)`
Hence, roots of the equation are `(-sqrt3)/(2)andx=(-sqrt3)/(2)`