Find the roots of the following quadratic equations by the factorisation method.
`2x^(2)+5/3x-2=0`

Explanation

(i) Given equation is `2x^(2)+(5)/(3)x-2=0`
`implies6x^(2)+5x-6=0`
`implies6x^(2)+(9-4)x-6=0`
`implies6x^(2)+9x-4x-6=0`
`implies3x(2x+3)-2(2x+3)=0`
`implies(3x-2)(2x+3)=0`
Now, `3x-2=0impliesx=(2)/(3)`
and `2x+3=0impliesx=(-3)/(2)`
Hence, `(2)/(3)and(-3)/(2)` are roots of the equation.
(ii) Given equation is `(2)/(5)x^(2)-x-(3)/(5)=0`
`implies2x^(2)-5x-3=0`
`implies2x^(2)-(6-1)x-3=0`
`implies2x^(2)-6x+x-3=0`
`implies2x(x-3)+(x-3)=0`
Now `2x+1=0impliesx=-(1)/(2)`
and `x-3=0impliesx=3`
Hence, roots of the equation are `3and-(1)/(2)`.
(iii) Given equation is `3sqrt2x^(2)-5x-sqrt2=0`
`implies3sqrt2x^(2)-(6-1))x-sqrt2=0`
`implies3sqrt2x^(2)-6x+x-sqrt2=0`
`implies3sqrt2x(x-sqrt2)+1(x-sqrt2)=0`
`implies(3sqrt2x+1)(x-sqrt2)=0`
Now, `3sqrt2x+1=0impliesx=-(1)/(3sqrt2)`
and `x-sqrt2=0impliesx=sqrt2`
Hence, `-(1)/(3sqrt2)andsqrt2` are roots of the equation.
(iv) Given equation is `3x^(2)+5sqrt5x-10=0`
`implies3x^(2)+6sqrt5x-sqrt5x-10=0`
`implies3x(x+2sqrt5)-sqrt5(x+2sqrt5)=0`
`implies(x+2sqrt5)(3x-sqrt5)=0`
Now `x+2sqrt5=0impliesx=-2sqrt5`
and `3x-sqrt5=0impliesx=(sqrt5)/(3)`
Hence, `-2sqrt5 and (sqrt5)/(3)` are roots of the equation.
(v) Given equation is `21x^(2)-2x+(1)/(21)=0`
`implies441x^(2)-42x+1=0`
`implies441x^(2)-21x-21x+1=0`
`implies21x(21x-1)-1(21x-1)=0`
`implies(21x-1)(21x-1)=0`
Now, `21x-1=0impliesx=(1)/(21)`
and `21x-1=0impliesx=(1)/(21)`
Hence, roots of the equation are `(1)/(21)and (1)/(21)`.