Check whether the following are Quadratic Equations.
(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)\(x^3 - 4x^2 - x + 1 = (x - 2)^3\)

(i)\((x + 1)^2 = 2(x - 3)\)
Using identity \((a + b)^2 = a^2 + 2ab + b^2\)
\( \Rightarrow x^2 + 1 + 2x = 2x - 6\)
\( \Rightarrow x^2 + 7 = 0\)

Hence it is a quadratic equation since the degree is 2.


(ii)\(x^2 - 2x = (-2) (3 - x)\)
\( \Rightarrow x^2 - 2x = -6 + 2x\)
\( \Rightarrow x^2 - 2x - 2x + 6 = 0\)
\( \Rightarrow x^2 - 4x + 6 = 0\)

Hence it is a quadratic equation with degree 2.


(iii)\((x - 2) (x + 1) = (x - 1) (x + 3)\)
\( \Rightarrow x^2 + x - 2x - 2 = x^2 + 3x - x - 3\)
\( \Rightarrow x^2 + x - 2x - 2 - (x^2 + 3x - x - 3) = 0\)
\( \Rightarrow x^2 - x^2 + x - 2x - 3x + x - 2 + 3 = 0\)
\( \Rightarrow -3x + 1 = 0\)

Hence it is a not quadratic equation since the degree is 1.


(iv)\((x - 3) (2x + 1) = x (x + 5)\)
\( \Rightarrow 2x^2 + x - 6x - 3 = x^2 + 5x\)
\( \Rightarrow 2x^2 + x - 6x - 3 - x^2 - 5x = 0\)
\( \Rightarrow x^2 - 10x - 3 = 0\)

Hence it is a quadratic equation since the degree is 2.


(v)\((2x - 1) (x - 3) = (x + 5) (x - 1)\)
\( \Rightarrow 2x^2 - x - 6x + 3 = x^2 - x + 5x - 5\)
\( \Rightarrow 2x^2 - 7x + 3 - x^2 + x - 5x + 5 = 0\)
\( \Rightarrow x^2 - 10x - 3 = 0\)

Hence it is a quadratic equation since the degree is 2.


(vi)\(x^2 + 3x + 1 = (x - 2)^2\)
Using identity \((a - b)^2 = a^2 - 2ab + b^2\)
\( \Rightarrow x^2 + 3x + 1 = x^2 - 4x + 4 \)
\( \Rightarrow x^2 + 3x + 1 - x^2 + 4x - 4 = 0\)
\( \Rightarrow 7x - 3 = 0\)

Hence it is a not quadratic equation since the degree is 1.


(vii)\((x + 2)^3 = 2x(x^2 - 1)\)
Using identity \((a + b)^3 = a^3 + b^3 + 3ab(a + b)\)
\( \Rightarrow x^3 + 2^3 + 3(x)(2)(x + 2) = 2x(x^2 - 1)\)
\( \Rightarrow x^3 + 8 + 6x(x + 2) = 2x^3 - 2x\)
\( \Rightarrow 2x^3 - 2x - x^3 - 8 - 6x^2 - 12x = 0\)
\( \Rightarrow x^3 - 6x^2 - 14x - 8 = 0\)

Hence it is a not quadratic equation since the degree is 3


(viii)\(x^3 - 4x^2 - x + 1 = (x - 2)^3\)
Using identity \((a - b)^3 = a^3 - b^3 - 3ab(a - b)\)
\( \Rightarrow x^3 - 4x^2 - x + 1 = x^3 - 2^3 - 3(x)(2)(x-2)\)
\( \Rightarrow -4x^2 - x + 1 = -8 - 6x^2 + 12x\)
\( \Rightarrow 2x^2 - 13x + 9 = 0\)

Hence it is a quadratic equation since the degree is 2