Check whether the following are quadratic equation
(i) `(x+1)^2=2(x-3)`
(ii) `x^2-2x=(-2)(3-x)`
(iii) `(x-2)(x+1)=(x-1)(x+3)`
(iv) `(x-3)(2x+1)=x(x+5)`
(v) `(2x-1)(x-3)=(x+5)(x-1)`
(vi) `x^2+3x+1=(x-2)^2`
(vii) `(x+2)^3=2x(x^2-1)`
(viii) `x^3-4x^2 – x + 1 = (x – 2)^3`

Explanation

(i) Given equation is
`(x+1)^(2)=2(x-3)`
`impliesx^(2)+2x+1=2x-6" "[:'(a+b)^(2)=a^(2)+2ab+b^(2)]`
`implies x^(2)+7=0`
which is of the from `ax^(2)+bx+c=0`, where b=0
Hence, it is a quadratic equation.
(ii) Given equation is
`x^(2)-2x=-2(3-x)`
`impliesx^(2)-2x=-6+2x`
`impliesx^(2)-4x+6=0`
which is of the form `ax^(2)+bx+c=0`
Hence, it is a quadratic equation.
(iii) Given equation is
`(x-2)(x+1)=(x-1)(x+3)`
`impliesx^(2)-2x+x-2=x^(2)-x+3x-3`
`impliesx^(2)-x-2=x^(2)+2x-3`
`impliesx^(2)-x-2-x^(2)-2x+3=0`
`implies-3x+1=0`
which is not of the form `ax^(2)+bx+c=0`
Hence, it is not a quadratic equation.
Given equation is (iv) Given equation is
`(x-3)(2x+1)=x(x+5)`
`implies2x^(2)-6x+x-3=x^(2)+5x`
`implies2x^(2)-5x-3-x^(2)-5x=0`
`x^(2)-10x-3=0`
which is of the form `ax^(2)+bx+c=0`
Hence, it is a quadratice equation.
(v) Given equation is
`(2x-1)(x-3)=(x+5)(x-1)`
`implies2x^(2)-x-6x+3=x^(2)+5x-x-5`
`implies2x^(2)-7x+3=x^(2)+4x-5`
`implies2x^(2)-7x+3-x^(2)-4x+5=0`
`impliesx^(2)=11x+8=0`
which is not of the form `ax^(2)+bx+c=0`
Hence, it is not a quadratic equation.
Hence, it is not a quadratic equation.
(vi) Given equation is
`x^(2)+3x+1=(x-2)^(2)`
`x^(2)+3x+1=x^(2)-4x+4`
`x^(2)+3x+1=x^(2)-4x+4" "[:'(a-b)^(2)=a^(2)-2ab+b^(2)]`
`x^(2)+3x+1-x^(2)+4x-4=0`
`7x-3=0`
which is not of the form `ax^(2)+bx+c=0`
Hence, it is not a quadratic equation.
(vii) Given equation is
`(x+2)^(3)=2x(x^(2)-1)`
`impliesx^(3)+8+6x(x+2)=2x^(3)-2x" "[(a+b)^(2)=a^(2)-2ab+b^(2)]`
`impliesx^(3)+8+6x^(2)+12x=2x^(3)-2x`
`implies2x^(2)+6x^(2)+14x+8=0`
which is not in the form `ax^(2)+bx+c=0`
Hence, it is not a quadratic equation.
(viii) Given equation is
`x^(3)-4x^(2)-x+1=(x-2)^(3)`
`impliesx^(3)+4x^(2)-x+1=x^(3)-8-6x(x-2)`
`[:'(a-b)^(3)=a^(3)-b^(3)-3ab(a-b)]`
`impliesx^(3)-4x^(2)-x+1=x^(3)-8-6x^(2)+12x`
`impliesx^(3)-4x^(2)-x+1-x^(3)+8+6x^(2)-12x=0`
`implies2x^(2)-13x+9=0`
which is of the form `ax^(2)+bx+c=0`
Hence, it is a quadratic equation.