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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) $2{x}^{2}-\sqrt{3x}+9=0$

(vi) ${x}^{2}-2x-\sqrt{x}-5=0$

(vii) 3x^{2} − 5x + 9 = x^{2} − 7x + 3

(viii) $x+\frac{1}{x}=1$

(ix) x^{2} − 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) 16x^{2} − 3 = (2x + 5) (5x − 3)

(xiv) (x + 2)^{3} = x^{3} − 4

(xv) x(x + 1) + 8 = (x + 2) (x − 2)

(i) x

(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) $2{x}^{2}-\sqrt{3x}+9=0$

(vi) ${x}^{2}-2x-\sqrt{x}-5=0$

(vii) 3x

(viii) $x+\frac{1}{x}=1$

(ix) x

(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) 16x

(xiv) (x + 2)

(xv) x(x + 1) + 8 = (x + 2) (x − 2)

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Solution

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,

$\frac{{x}^{4}+1}{{x}^{2}}=5\phantom{\rule{0ex}{0ex}}\Rightarrow {x}^{4}+1=5{x}^{2}\phantom{\rule{0ex}{0ex}}\Rightarrow {x}^{4}-5{x}^{2}+1=0$

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

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