Question

Find a quadratic polynomial, the sum and product of whose zeroes are
$\sqrt{2}$
and
$- 3 \2$
respectively. Also, find its zeroes.​

Answer 1

${\Large{\underline{\underline{\bf{ \red{Question:}}}}}}$

Find a quadratic polynomial, the sum and product of whose zeroes are $\sqrt{2}$ and $\cfrac{-3}{2}$ respectively. Also, find its zeroes.

${\Large{\underline{\underline{\bf{ \red{Answer :}}}}}}$

Given :-

• Sum of the zeroes = $\sqrt{2}$
• Product of the zeroes = $\cfrac{-3}{2}$

To find :-

• Quadratic Polynomial

● We know that quadratic polynomial in form of :-

• ax² + bx + c

—————————————————————————

● Also Sum of the zeroes is given by

${\underline{\boxed{\sf{Sum \: of \: the \: Zeroes = \frac{ -(Coefficient \: of \: x) }{Coefficient \: of \: x^{2} } = \frac{( - b)}{a} }}}}$

${\sf{Sum \: of \: the \: Zeroes = - \cfrac{b}{a} }}$

${\sf{Sum \: of \: the \: Zeroes = \cfrac{ - \sqrt{2} }{1} }}$

● And Product of the Zeroes is given by

${\underline{\boxed{\sf{Product \: of \: the \: Zeroes = \frac{ Constant \: Term}{ Coefficient \: of \: x^{2}} = \frac{c}{a} }}}}$

Assuming that a = 1

${\sf{Product \: of \: the \: Zeroes \implies \cfrac{c}{1} = \cfrac{ - 3}{2} }}$

${\sf{Product \: of \: t he \: Zeroes \implies \: c \: = \cfrac{ - 3}{2} }}$

—————————————————————————

● Here values of a, b & c are ,

• a = 1
• b = - √2
• c = $\cfrac{-3}{2}$

■ So, Quadratic Polynomial made is

${\large{\bf{\implies {ax^{2} + bx + c }}}}$

${\large{\sf{\implies {1x^{2} + (- \sqrt{2} x )+ \bigg ( - \cfrac{ 3}{2} \bigg) }}}}$

${\large{\sf{\implies {x^{2} - \sqrt{2} x - \cfrac{3}{2} }}}}$

Multiplying 2 by whole polynomial, then we get.

${\large{\sf{\implies2 \bigg( {x^{2} - \sqrt{2} x - \cfrac{3}{2} \: \: \bigg ) }}}}$

${ \underline{ \boxed{{\large{\bf{ \pink{\implies {2x^{2} - 2 \sqrt{2} x - 3 \: \: }}}}}}}}$

—————————————————————————

■ Zeroes of the polynomial are :-

It can be find by given formula :-

$\large{\underline{\boxed{\implies{\bf{\cfrac{-b \pm \sqrt{b^{2}-4ac}}{2a}}}}}}$

● First Zeroes of polynomial is

$\large{\implies{\bf{\cfrac{-b + \sqrt{b^{2}-4ac}}{2a}}}}$

$\large{\implies{\sf{\cfrac{-(-2\sqrt{2}) + \sqrt{(-2\sqrt{2})^{2}-4(2)(-3)}}{2\times2}}}}$

$\large{\implies{\sf{\cfrac{2\sqrt{2} + \sqrt{8 + 24} }{4} }}}$

$\large{\implies{\sf{\cfrac{\sqrt{2} + \sqrt{8 + 24} }{2} }}}$

$\large{\implies{\sf{\cfrac{\sqrt{2} + \sqrt{32} }{2} }}}$

$\large{\implies{\underline{\boxed{\pink{\sf{\cfrac{3\sqrt{2} }{2} }}}}}}$

● Second Zeroes of the Polynomial is

$\large{\implies{\bf{\cfrac{-b - \sqrt{b^{2}-4ac}}{2a}}}}$

$\large{\implies{\sf{\cfrac{-(-2\sqrt{2}) - \sqrt{(-2\sqrt{2})^{2}-4(2)(-3)}}{2\times2}}}}$

$\large{\implies{\sf{\cfrac{2\sqrt{2} - \sqrt{8 + 24} }{4} }}}$

$\large{\implies{\sf{\cfrac{\sqrt{2} - \sqrt{8 + 24} }{2} }}}$

$\large{\implies{\sf{\cfrac{\sqrt{2} - \sqrt{32} }{2} }}}$

$\large{\implies{\underline{\boxed{\pink{\sf{-\cfrac{\sqrt{2} }{2} }}}}}}$