Question

Find a quadratic polynomial whose sum ans product of zeroes are -1/4 and 1/4 respectively

Answer 1

Given : The sum and product of zeroes of a Quadratic Polynomial are -1/4 and 1/4 , respectively .

Exigency To Find : The Quadratic Polynomial .

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⠀⠀⠀⠀⠀Finding Quadratic polynomial :

$\dag\:\:\sf{ As,\:We\:know\:that\::}\\\\ \qquad\bigstar\:\:\bf Quadratic\:Polynomial\::$

$\qquad \dag\:\:\bigg\lgroup \sf{ Quadratic \:Polynomial \: =\:\: x^2 - ( \: \alpha \:+ \beta \:)x \: + ( \:\alpha \times \:\beta \:) \:\:=0\:}\bigg\rgroup$

⠀⠀⠀⠀⠀Here , $\: \alpha + \beta \:$ is the sum of Zeroes & $\: \alpha \times \beta \:$ is the product of Zeroes .

⠀$\qquad \dashrightarrow \:\sf Quadratic \:Polynomial \: =\:\: x^2 - ( \: \alpha \:+ \beta \:) x\: + ( \:\alpha \times \:\beta \:) \:=\:0\:$

$\qquad \dashrightarrow \:\sf \:\: x^2 - ( \: \alpha \:+ \beta \:)x \: + ( \:\alpha \times \:\beta \:) \:=\:0\:$

⠀⠀⠀⠀⠀⠀$\underline {\boldsymbol{\star\:Now \: By \: Substituting \: the \: known \: Values \::}}$

$\qquad \dashrightarrow \:\sf \:\: x^2 - ( \: \alpha \:+ \beta \:)x \: + ( \:\alpha \times \:\beta \:) \:=\:0\:$

$\qquad \dashrightarrow \:\sf \:\: x^2 - \bigg( \: -\dfrac{1}{4} \:\bigg)x \: + \bigg( \:\dfrac{1}{4} \:\bigg) \:=\:0\:$⠀⠀

$\qquad \dashrightarrow \:\sf \:\: x^2 + \: \dfrac{1}{4} \:x \: + \:\dfrac{1}{4} \:\:=\:0\:$⠀⠀

$\qquad \dashrightarrow \:\sf \:\: \: \dfrac{4x^2 \:\:\:+ \: 1x \:\: + \:\: 1 }{4} \: \: \:\:=\:0\:$⠀⠀

$\qquad \dashrightarrow \:\sf \:\: \: \dfrac{4x^2 \:\:\:+ \: x \:\: + \:\: 1 }{4} \: \: \:\:=\:0\:$⠀⠀

$\qquad \dashrightarrow \:\sf \:\: \: 4x^2 \:\:\:+ \: x \:\: + \:\: 1 \: \: \:\:=\:0\:\times 4$⠀⠀

$\qquad \dashrightarrow \:\sf \:\: \: 4x^2 \:\:\:+ \: x \:\: + \:\: 1 \: \: \:\:=\:0\:$⠀⠀

$\qquad \dashrightarrow \pmb{\underline{\purple{\: Quadratic\:Polynomial\: =\:4x^2 \:\: +\: x\: +\:1\: }} }\:\;\bigstar$

Therefore,

⠀⠀⠀⠀⠀$\therefore {\underline{ \mathrm {\:Hence,\:The \:Quadratic \:Polynomial \:is\:\bf{ \:4x^2 \:\: +\: x\: + \: 1\:\: }}.}}$

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⠀⠀$\large {\boxed{\sf{\mid{\overline {\underline {\star More\:To\:know\::}}}\mid}}}$

$\qquad \qquad \boxed {\begin{array}{cc} \bf{\underline {\bigstar\:\: For \: a \:Quadratic \:Polynomial \::}}\\\\ \sf{ Whose \:\:zeroes \:\:are\:\:\alpha \:\&\;\: \beta\:\:} \\\\ 1)\:\: \alpha + \beta \: =\:\dfrac{-b}{a} \quad \bigg\lgroup \bf Sum\:of\;Zeroes \bigg\rgroup \\\\ 2)\:\: \alpha \times \beta \: =\:\dfrac{c}{a} \quad \bigg\lgroup \bf Product \:of\;Zeroes \bigg\rgroup \\\\ \end{array}}$

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Answer 2

Answer:

2. $4x ^{2} + x + 1$