Check whether the following are quadratic equations: (i) (x + 1) 2 = 2(x - 3) (ii) x2 - 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) x2 + 3x + 1 = (x - 2) 2 (vii) (x + 2) 2 = 2x (x2 - 1) (viii) x3 - 4x2 - x + 1 = (x - 2) 3

i)(x + 1)2 = 2(x – 3)

By using the identity (a+b)= a2+2ab+b2

⇒ x2 + 2x + 1 = 2x – 6

⇒ x2 + 7 = 0

Since the above equation is in the form of ax2 + bx + c = 0.

Therefore, the given equation is quadratic equation.

(ii) x2 – 2x = (–2) (3 – x)

By using the identity (a+b)= a2+2ab+b2

⇒ x–2x = -6 + 2x

⇒ x– 4x + 6 = 0

Since the above equation is in the form of ax2 + bx + c = 0.

Therefore, the given equation is quadratic equation.

(iii) (x – 2)(x + 1) = (x – 1)(x + 3)

By using the identity (a+b)= a2+2ab+b2

⇒ x– x – 2 = x+ 2x – 3

⇒ 3x – 1 = 0

Since the above equation is not in the form of ax2 + bx + c = 0.

Therefore, the given equation is not a quadratic equation.

(iv)(x – 3)(2x +1) = x(x + 5)

By using the identity (a+b)2=a2+2ab+b2

⇒ 2x– 5x – 3 = x+ 5x

⇒ x– 10x – 3 = 0

Since the above equation is in the form of ax2 + bx + c = 0.

Therefore, the given equation is quadratic equation.

(v)(2x – 1)(x – 3) = (x + 5)(x – 1)

By using the identity (a+b)2=a2+2ab+b2

⇒ 2x– 7x + 3 = x+ 4x – 5

⇒ x– 11x + 8 = 0

Since the above equation is in the form of ax2 + bx + c = 0.

Therefore, the given equation is quadratic equation.

(vi) x2 + 3x + 1 = (x – 2)2

By using the formula for (a+b)2=a2+2ab+b2

⇒ x2 + 3x + 1 = x2 + 4 – 4x

⇒ 7x – 3 = 0

Since the above equation is not in the form of ax2 + bx + c = 0.

Therefore, the given equation is not a quadratic equation.

(vii) (x + 2)3 = 2x(x2 – 1)

By using the formula for (a+b)= a2+2ab+b2

⇒ x3 + 8 + x2 + 12x = 2x3 – 2x

⇒ x3 + 14x – 6x2 – 8 = 0

Since the above equation is not in the form of ax2 + bx + c = 0.

Therefore, the given equation is not a quadratic equation.

(viii) x3 – 4x2 – x + 1 = (x – 2)3

By using the formula for (a+b)= a2+2ab+b2

⇒ x3 – 4x2 – x + 1 = x3 – 8 – 6x + 12x

⇒ 2x2 – 13x + 9 = 0

Since the above equation is in the form of ax2 + bx + c = 0.

Therefore, the given equation is quadratic equation.

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