Question

Which of the following are quadratic equations?

(i) \(x^{2}-3x=0\)

(ii) \((x+\frac{1}{x})^{2}=3(x+\frac{1}{x})+4\) 

(iii) \((2x+1)(3x+2)=6(x-1)(x-2)\) 

(iv) \(x+\frac{1}{x}=x^{2},x\neq0\) 

(v) \(16x^{2}-3=(2x+5)(5x-3)\) 

(vi) \((x+2)^{3}=x^{3}-4\) 

(vii) \(x(x+1)+8=(x+2)(x-2)\)

Answer 1

(i) \(x^{2}-3x=0\) 

It is a quadratic equation.

(ii) \((x+\frac{1}{x})^{2}=3(x+\frac{1}{x})+4\) 

\(\Rightarrow x^{2}+\frac{1}{x^{2}}+2=3x+\frac{3}{x}+4\) 

⇒ x⁴ + 1 + 2x² = 3x³ + 3x + 4x²

It is not a quadratic equation.

(iii) \((2x+1)(3x+2)=6(x-1)(x-2)\) 

⇒ 6x² + 7x + 2 = 6x² – 18x + 12

⇒ 25x = 10

It is not a quadratic equation.

(iv) \(x+\frac{1}{x}=x^{2},x\neq0\) 

⇒ x² + 1 = x³

It is not a quadratic equation.

(v) \(16x^{2}-3=(2x+5)(5x-3)\) 

⇒ 16x² – 3 = 10x² + 19x – 15

⇒ 6x² – 19x + 12 = 0

It is a quadratic equation.

(vi) \((x+2)^{3}=x^{3}-4\) 

⇒ x³ + 8 + 6x² + 12x = x³ – 4

⇒ 6x² + 12x + 12 = 0

It is a quadratic equation.

(vii) \(x(x+1)+8=(x+2)(x-2)\) 

⇒ x² + x + 8 = x² – 4

⇒ x = - 12

It is not a quadratic equation.