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

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