Question

Find a quadratic polynomial , the sum and product of whose zeroes are 0 and - root2, respectively

Answer 1

$\blue{\bold{\underline{\underline{Answer:}}}}$

$\green{\tt{\therefore{Equation=x^{2}-\sqrt{2}=0}}}$

$\orange{\bold{\underline{\underline{Step-by-step\:explanation:}}}}$

$\green{\underline \bold{Given:}} \\ \tt: \implies Sum \: of \: zeroes = 0 \\ \\ \tt: \implies Product \: of \: zeroes = - \sqrt{2} \\ \\ \red{\underline \bold{To \: Find:}} \\ \tt: \implies Quadratic \:equation = ?$

• According to given question :

$\circ \: \tt{Let \: \alpha \: and \: \beta \: be \: the \: zeroes} \\ \\ \tt \circ \: sum \: of \: zeroes = 0 \\ \\ \tt: \implies \alpha + \beta = 0 \\ \\ \tt \circ \: product \: of \: zeroes = - \sqrt{2} \\ \\ \tt: \implies \alpha \beta = - \sqrt{2} \\ \: \\ \bold{As \: we \: know \: that} \\ \tt: \implies {x}^{2} - (sum \: of \: zeroes)x + (product \: of \: zeroes) = 0\\ \\ \tt: \implies {x}^{2} - ( \alpha + \beta )x + \alpha \beta = 0 \\ \\ \text{Putting \: given \: values} \\ \tt: \implies {x}^{2} - 0 \times x + ( - \sqrt{2} ) = 0 \\ \\ \green{\tt: \implies {x}^{2} - \sqrt{2} = 0}$

Answer 2

AnswEr:-

Quadratic polynomial = x² - √2

$\rule{200}{1}$

Given:-

☛ Sum of zeroes = 0

☛ Product of zeroes = -√2

To find:-

☛ Quadratic polynomial = ?

Solution:-

Let the zeroes of polynomial be α & β

The standard form of an quadratic polynomial is :-

↠x² - (Sum of zeroes)x + (Product of zeroes)

↠ x² - (α + β)x + αβ

Here,sum of zeroes = 0

∴ α + β = 0

And,product of zeroes = -√2

∴ αβ = -√2

$\rule{200}{1}$

⋆ QUADRATIC POLYNOMIAL:-

⇒ x² - (0)x + (-√2)

⇒ x² - 0x - √2

⇒ x² - √2

∴Quadratic polynomial = x² - √2