Question

Which of the following are quadratic equations?

(i) x² + 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) 3x² − 5x + 9 = x² − 7x + 3
(viii) $x+\frac{1}{x}=1$
(ix) x² − 3x = 0
(x) ${\left(x+\frac{1}{x}\right)}^2=3\left(x+\frac{1}{x}\right)+4$
(xi) (2x + 1) (3x + 2) = 6(x − 1) (x − 2)
(xii) $x+\frac{1}{x}=x^2,x\ne 0$
(xiii) 16x² − 3 = (2x + 5) (5x − 3)
(xiv) (x + 2)³ = x³ − 4
(xv) x(x + 1) + 8 = (x + 2) (x − 2)

Answer 1

We are given the following algebraic expressions and are asked to find out which one is quadratic.

(i) Here it has been given that,

Now, the above equation clearly represents a quadratic equation of the form , where , and .

Hence, the above equation is a quadratic equation.

(ii) Here it has been given that,

Now, solving the above equation further we get,

Now, the above equation clearly represents a quadratic equation of the form , where , and .

Hence, the above equation is a quadratic equation.

(iii) Here it has been given that,

Now, solving the above equation further we get,

$$\begin{gathered} \frac{x^4+1}{x^2}=5 \\ \Rightarrow x^4+1=5x^2 \\ \Rightarrow x^4-5x^2+1=0 \end{gathered}$$

Now, the above equation clearly does not represent a quadratic equation of the form , because $x^4-5x^2+1$ is a polynomial of degree 4.

Hence, the above equation is not a quadratic equation.

(iv) Here it has been given that,

Now, solving the above equation further we get,

Now, the above equation clearly does not represent a quadratic equation of the form , because is a polynomial of degree 3.

Hence, the above equation is not a quadratic equation.

(v) Here it has been given that,

Now, the above equation clearly does not represent a quadratic equation of the form, because contains a term , where is not an integer.

Hence, the above equation is not a quadratic equation.

(vi) Here it has been given that,

Now, as we can see the above equation clearly does not represent a quadratic equation of the form, because contains an extra term, where is not an integer.

Hence, the above equation is not a quadratic equation.

(vii) Here it has been given that,

Now, after solving the above equation further we get,

Now as we can see, the above equation clearly represents a quadratic equation of the form , where , and .

Hence, the above equation is a quadratic equation.

(viii) Here it has been given that,

Now, solving the above equation further we get,

Now as we can see, the above equation clearly represents a quadratic equation of the form , where , and .

Hence, the above equation is a quadratic equation.

(ix) Here it has been given that,

Now as we can see, the above equation clearly represents a quadratic equation of the form, where, and.

Hence, the above equation is a quadratic equation.

(x) Here it has been given that,

Now, solving the above equation further we get,

Now as we can see, the above equation clearly does not represent a quadratic equation of the form , because is a polynomial having a degree of 4 which is never present in a quadratic polynomial.

Hence, the above equation is not a quadratic equation.

(xi) Here it has been given that,

Now, after solving the above equation further we get,

Now as we can see, the above equation clearly does not represent a quadratic equation of the form , because is a linear equation.

Hence, the above equation is not a quadratic equation.

(xii) Here it has been given that,

Now, solving the above equation further we get,

Now as we can see, the above equation clearly does not represent a quadratic equation of the form , because is a polynomial having a degree of 3 which is never present in a quadratic polynomial.

Hence, the above equation is not a quadratic equation.

(xiii) Here it has been given that,

Now, after solving the above equation further we get,

Now as we can see, the above equation clearly represents a quadratic equation of the form , where , and.

Hence, the above equation is a quadratic equation.

(xiv) Here it has been given that,

Now, after solving the above equation further we get,

Now as we can see, the above equation clearly represents a quadratic equation of the form , where , and .

Hence, the above equation is a quadratic equation.

(xv) Here it has been given that,

Now, solving the above equation further we get,

Now as we can see, the above equation clearly does not represent a quadratic equation of the form , because is a linear equation which does not have a term in it.

Hence, the above equation is not a quadratic equation.