Check whether the following are quadratic equations:
(i) `(2x-1)(x-3)=(x+4)(x-2)" "(ii)" "(x+2)^(3)=2x(x^(2)-1)`
(iii) `(x+1)^(3)=x^(3)+x+6" "(iv)" "x(x+3)+6=(x+2)(x-2)`

Explanation

We have
`(i)" "(2x-1)(x-3)=(x+4)(x-2)`
`implies" "2x^(2)-7x+3=x^(2)+2x-8impliesx^(2)-9x+11=0.`
This is of the form `ax^(2)+bx+c=0`, where `a=1,b=-9` and `c=11`.
Hence, the given equation is a quadratic equation.
(ii) `(x+2)^(3)=2x(x^(2)-1)`
`implies" "x^(3)+8+6x(x+2)=2x^(3)-2x`
`implies" "x^(3)+6x^(2)+12x+8=2x^(3)-2x`
`implies" "x^(3)-6x^(2)-14x-8=0.`
This is not of the form `ax^(2)+bx+c=0.`
Hence, the given equation is not a quadratic equation.
(iii) `(x+1)^(3)=x^(3)+x+6`
`implies" "x^(3)+1+3x(x+1)=x^(3)+x+6`
`implies" "3x^(2)+2x-5=0.`
This is of the form `ax^(2)+bx+c=0`, where `a=3,b=2` and `c=-5.`
Hence, the given equation is a quadratic equation.
(iv)` x(x+3)+6=(x+2)(x-2)`
`implies" "x^(2)+3x+6=x^(2)-4`
`implies" "3x+10=0.`
This is not of the form `ax^(2)+bx+c=0.`
Hence, the given equation is not a quadratic equation.