Question

7.Find a quadratic polynomial, the sum and product of whos zeroes are v2 and -3/2 respectively.
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Answer 1

Answer:

Given :-

• The sum and product of zeroes are √2 and - 3/2 respectively.

To Find :-

• What is the quadratic polynomial.

Formula Used :-

$\clubsuit$ Quadratic Polynomial Formula :

$\footnotesize\mapsto \sf\boxed{\bold{\pink{x^2 - (Sum\: Of\: Roots)x + (Product\: Of\: Roots)}}}$

Solution :-

Given :

❒ Sum Of Roots (α + β) = √2

❒ Product Of Roots (αβ) = - 3/2

According to the question by using the formula we get,

$\longrightarrow \sf\bold{\purple{x^2 - (\alpha + \beta)x + (\alpha\beta)}}$

$\longrightarrow \sf x^2 - (\sqrt{2})x + \bigg(- \dfrac{3}{2}\bigg)$

$\longrightarrow \sf\bold{\red{x^2 - \sqrt{2}x - \dfrac{3}{2}}}$

${\small{\bold{\underline{\therefore\: The\: required\: quadratic\: polynomial\: is\: x^2 - \sqrt{2}x - \dfrac{3}{2}\: .}}}}$

Answer 2

$\: \: \: \: \boxed{ \bf \: {x}^{2} - \sqrt{2} \: x - \frac{3}{2} }$

Explanation :

$\bf \leadsto \: given \\ \\ \small{ \rm \to \: sum \: of \: zeroes \: of \: a \: quadratic \: polynomial : } \\ \: \: \: \: \: \rm \: \alpha + \beta = \sqrt{2} \\ \\ \small{ \rm \to \: product \: of \: zeroes \: of \: a \: quadratic \: polynomial : } \\ \: \rm \: \alpha \beta = - \frac{3}{2}$

→ We have to find that quadratic polynomial.

Solution :

$\pink{ \boxed{\boxed{\begin{array}{cc} \underline{ \bf \: we \: know \: that} \\ \\ \sf \to \: general \: formula \: of \: a \: quadratic \: \\ \sf \: polynomial \: \: is \: : \\ \\ \blue{ \boxed{ \bf \: {x}^{2} - ( \alpha + \beta )x - \alpha \beta }} \\ \\ \bf \: here \\ \\ \bf \: \alpha + \beta = sum \: of \: zeroes \\ \\ \bf \: \alpha \beta = product \: of \: zeroes \end{array}}}}$

Now, according to the question

The quadratic polynomial is :

${ \boxed{\boxed{\begin{array}{cc} \bf \: {x}^{2} - ( \alpha + \beta)x + \alpha \beta \\ \\ \bf \implies \: {x}^{2} - \sqrt{2} \: x + ( - \frac{3}{2}) \\ \\ \bf \implies \: {x}^{2} - \sqrt{2} \: x - \frac{3}{2} \end{array}}}}$