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1. 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)3 = 2x (x2 – 1) (viii) x3 – 4x2 – x + 1 = (x – 2)3

1. 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)3 = 2x (x2 – 1)

(viii) x3 – 4x2 – x + 1 = (x – 2)3

Grade:12th pass

1 Answers

Pawan Prajapati
askIITians Faculty 8740 Points
4 months ago
(i) Given, (x + 1)2 = 2(x – 3) By using the formula for (a+b)2 = 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) Given, x2 – 2x = (–2) (3 – x) By using the formula for (a+b)2 = a2+2ab+b2 ⇒ x2 – 2x = -6 + 2x ⇒ x2 – 4x + 6 = 0 Since the above equation is in the form of ax2 + bx + c = 0. Therefore, the given equation is quadratic equation. (iii) Given, (x – 2)(x + 1) = (x – 1)(x + 3) By using the formula for (a+b)2 = a2+2ab+b2 ⇒ x2 – x – 2 = x2 + 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) Given, (x – 3)(2x +1) = x(x + 5) By using the formula for (a+b)2=a2+2ab+b2 ⇒ 2x2 – 5x – 3 = x2 + 5x ⇒ x2 – 10x – 3 = 0 Since the above equation is in the form of ax2 + bx + c = 0. Therefore, the given equation is quadratic equation. (v) Given, (2x – 1)(x – 3) = (x + 5)(x – 1) By using the formula for (a+b)2=a2+2ab+b2 ⇒ 2x2 – 7x + 3 = x2 + 4x – 5 ⇒ x2 – 11x + 8 = 0 Since the above equation is in the form of ax2 + bx + c = 0. Therefore, the given equation is quadratic equation. (vi) Given, 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) Given, (x + 2)3 = 2x(x2 – 1) By using the formula for (a+b)2 = 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) Given, x3 – 4x2 – x + 1 = (x – 2)3 By using the formula for (a+b)2 = a2+2ab+b2 ⇒ x3 – 4x2 – x + 1 = x3 – 8 – 6x2 + 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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