Question

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) \(x^3 – 4x^2 – x + 1 = (x – 2)^3\)

Answer 1

(i) \((x + 1)^2 = 2(x – 3) ⇒ x^2 + 2x + 1 = 2x -6 ⇒ x^2 +7 =0\)
It is of the form \(ax^2 + bx +c =0.\)

Hence, the given equation is a quadratic equation.

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(ii) \(x^2 – 2x = (–2) (3 – x) ⇒ x^2 -2x = -6 + 2x ⇒ x^2 -4x +6\)
It is of the form \(ax^2 + bx +c =0.\)

Hence, the given equation is a quadratic equation.

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(iii) \((x – 2)(x + 1) = (x – 1)(x + 3) ⇒ x^2 -x-2 = x^2 +2x -3 ⇒ 3x-1\)
It is not of the form \(ax^2 + bx +c =0.\)

Hence, the given equation is not a quadratic equation.

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(iv) \((x – 3)(2x +1) = x(x + 5) ⇒ 2x^2 -5x -3 = x^2 +5x ⇒ x^2 -10x -3\)
It is of the form \(ax^2 + bx +c =0.\)

Hence, the given equation is a quadratic equation.

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(v) \((2x – 1)(x – 3) = (x + 5)(x – 1) ⇒ 2x^2 -7x +3 = x^2 +4x -5 ⇒ x^2-11x +8 =0\)
It is of the form \(ax^2 + bx +c =0.\)

Hence, the given equation is a quadratic equation.

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(vi) \(x^2 + 3x + 1 = (x – 2)^2 ⇒ x^2 +3x +1 = x^2 +4 -4x ⇒7x -3 =0\)
It is not of the form \(ax^2 + bx +c =0.\)

Hence, the given equation is not a quadratic equation.

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(vii) \((x + 2)^3 = 2x (x^2 – 1) ⇒x^3 +8 +6x^2 +12x ⇒ 2x^3 -2x ⇒ x^2 -14x -6x^2 -8=0\)
It is of the form \(ax^2 + bx +c =0.\)

Hence, the given equation is a quadratic equation.

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(viii) \(x^3 – 4x^2 – x + 1 = (x – 2)^3 ⇒ x^3 -4x^2 -x +1 = x^3 -8-6x^2 +12x ⇒ 2x^2 -13x +9\)
It is of the form \(ax^2 + bx +c =0.\)

Hence, the given equation is a quadratic equation.