Question 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
Answer:
(i) (x + 1)2 = 2(x – 3) ⇒x2 + 2x + 1 = 2x – 6 ⇒ x2 + 7 = 0
It is of the form ax2 + bx +c = 0.
Hence, the given equation is a quadratic equation.
(ii) x2 – 2x = (–2) (3 – x) ⇒ x2 – 2x = –6 + 2x ⇒ x2 – 4x + 6 = 0
It is of the form ax2 + bx + c = 0
Hence, the given equation is a quadratic equation.
(iii) (x – 2) (x + 1) = (x – 1) (x + 3) ⇒ x2 – x – 2 = x2 + 2x – 3 ⇒ 3x – 1 = 0
It is not of the form ax2 + bx + c = 0.
Hence, the given equation is not a quadratic equation.
(iv) (x – 3) (2x + 1) = x(x + 5) ⇒ 2x2 – 5x – 3 = x2 + 5x ⇒ x2 – 10x – 3 = 0
It is of the form ax2 + bx + c = 0.
Hence, the given equation is a quadratic equation.
(v) (2x −1) (x – 3) = (x + 5) (x – 1) ⇒ 2x2 – 7x + 3 = x2 + 4x – 5 ⇒ x2 −11x + 8 = 0, it is of the form ax2 + bx + c = 0.
Hence, the given equation is a quadratic equation.
(vi) x2 + 3x + 1 = (x – 2)2 ⇒ x2 + 3x + 1 = x2 + 4 – 4x ⇒ 7x – 3 = 0
It is not of the form ax2 + bx + c = 0.
Hence, the given equation is not a quadratic equation.
(vii) (x + 2)3 = 2x (x2 – 1) ⇒ x3 + 8 + 6x2 + 12x = 2x3 – 2x ⇒ x3 – 14x – 6x2 – 8 = 0 . It is not of the form ax2 + bx + c = 0.
Hence, the given equation is not a quadratic equation.
(viii) x3 – 4x3 – x + 1 = (x – 2)3 ⇒ x3 – 4x2 – x + 1 = x3 – 8 – 6x2 + 12x ⇒ 2x2 – 13x + 9 = 0
It is of the form ax2 + bx + c = 0.
Hence, the given equation is a quadratic equation.
Latest Govt Job & Exam Updates: