Question

what is the quadratic polynomial the sum of whose zeroes are -3 by 2 and the product of the zeros is -1​

Answer 1

Given

⇒Sum of the zeroes(α + β) = -3/2

⇒Product of Zeroes (αβ) = -1

To form a Quadratic  Polynomial , we use this

⇒x² - (α + β)x + αβ = 0

Now Put the value

⇒x² - (-3/2)x + (-1) = 0

⇒x² + 3x/2 - 1 = 0

Now Taking Lcm , we get

⇒(2x² + 3x - 2)/2 = 0

⇒2x² + 3x - 2 = 0

Answer

⇒2x² + 3x - 2 = 0

                                                             

More Information

⇒Root of Quadratic x =  {-b±√(b²-4ac)}/2a

⇒To Form a Quadratic Equation

x² - (α + β)x + αβ = 0

⇒Discriminant(D) = b² - 4ac

⇒Sum of Roots(α+β) = -b/a

⇒Product(αβ) = c/a

Answer 2

Given : The sum of whose zeroes are $\dfrac{-3}{2}$ and the product of the zeros is -1

Exigency to find : The Quadratic Polynomial.

⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━⠀

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

$\qquad \dag\:\:\bigg\lgroup \sf{ x^2 - (sum\:of\:zeroes)x + Product \:of\:zeroes \:\: }\bigg\rgroup$

⠀⠀⠀⠀⠀Here sum of zeroes are $\bf\dfrac{-3}{2}$ and the product of the zeros is -1 .

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

$\qquad :\implies \sf x^2 - \bigg( \dfrac{-3}{2}\bigg) x + (-1 ) = 0$

$\qquad :\implies \sf x^2 + \dfrac{3}{2} x - 1 = 0$

$\qquad :\implies \sf \dfrac{2x^2 + 3x - 2 }{2} = 0$

$\qquad :\implies \sf 2x^2 + 3x - 2 = 0 \times 2$

$\qquad :\implies \sf 2x^2 + 3x - 2 = 0$

$\qquad \longmapsto \frak{\underline{\purple{\:\big( 2x^2 + 3x - 2\big) \quad \longrightarrow Quadratic\:Polynomial\:}} }\bigstar$

Therefore,

⠀⠀⠀⠀⠀$\therefore {\underline{ \mathrm {\:The\:Required \:Quadratic \:Polynomial \: \:is\:\bf{2x^2+3x-2}}}}$

$\rule{300}{1.5}$

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

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

⠀⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━