Question

Which of the following are quadratic equations?

(i) \(x^{2}+6x-4=0\) 

(ii) \(\sqrt{3}x^{2}-2x+\frac{1}{2}=0\) 

(iii) \(x^{2}+\frac{1}{x^{2}}=5\) 

(iv) \(x-\frac{3}{x}=x^{2}\)

(v) \(2x^{2}-\sqrt{3x}+9=0\)

(vi) \(x^{2}-2x-\sqrt{x}-5=0\)

(vii) \(3x^{2}-5x+9=x^{2}-7x+3\) 

(viii) \(x+\frac{1}{x}=1\) 

Answer 1

A polynomial equation is a quadratic equation, if it is of the form ax² + bx + c = 0 such that a ≠ 0

(i) \(x^{2}+6x-4=0\)

It is a quadratic equation.

(ii) \(\sqrt{3}x^{2}-2x+\frac{1}{2}\)

It is a quadratic equation.

(iii) \(x^{2}+\frac{1}{x^{2}}=5\) 

⇒ x⁴ -5x² + 1 = 0

It is not a quadratic equation as the highest power of x is ‘4’

(iv) \(x-\frac{3}{x}=x^{2}\) 

⇒ x² – 3 = x³

It is not a quadratic equation.

(v) \(2x^{2}-\sqrt{3x}+9=0\)

It is not a quadratic equation as √x is present instead of ‘x’.

(vi) \(x^{2}-2x-\sqrt{x}-5=0\) 

It is not a quadratic equation as an additional √x term is present.

(vii) \(3x^{2}-5x+9=x^{2}-7x+3\) 

⇒ 2x² + 2x + 6 = 0

It is a quadratic equation.

(viii) \(x+\frac{1}{x}=1\)

⇒ x² + 1 – x = 0

It is a quadratic equation.