Which of the following is not a quadratic equation ? (i) `(sqrt2x+sqrt3)^2=3x^2-5x`

Options

A. `2(x-1)^(2)=4x^(2)-2x+1`
B. `2x-x^(2)=x^(2)+5`
C. `(sqrt(2)x+sqrt(3))^(2)=3x^(2)-5x`
D. `(x^(2)+2x)^(2)=x^(4)+3+4x^(2)`

Explanation

(a) Given that `2(x-1)^(2)=4x^(2)-2x+1`
`implies 2(x^(2)+1-2x)=4x^(2)-2x+1`
`implies 2x^(2)+2-4x=4x^(2)-2x+1`
`= 2x^(2)+2x-1=0`
which represents a quadratic equation because it has the quadratic form
`ax^(2)+bx+c=0, a!=0`
(b) Given that `2x-x^(2)=x^(2)+5`
`= 2x^(2)-2x+5=0`
which also represents a quadratic equation because it has the quadratic form
`ax^(2)+bx+c=0,a!=0`
(c) Given that `(sqrt(2).x+sqrt(3))^(2)=3x^(2)-5x`
`implies 2.x^(2)+3+2sqrt(6).x=3x^(2)-5x`
`implies x^(2)-(52sqrt(5))x-3=0`
which also represents a quadratic equation because it has the quadratic form
`ax^(2)+bx+c=0, a!=0`
(d) Given that `(x^(2)+2x)^(2)=x^(4)+3x+4x^(2)`
`implies x^(4)+4x^(2)+4x^(3)+3+4x^(2)`
`implies 4x^(3)-3=0`
which is not of the form `ax^(2)+bx+c,a!=0`. Thus, the equation is not quadratic. This is a cubic equation.