Question

Find the discriminant of the quadratic equation 2x square -4x + 3 =0 and the hence find the natural of its roofs

Answer 1

EXPLANATION.

Quadratic equation.

⇒ 2x² - 4x + 3 = 0.

As we know that,

⇒ D = Discriminant Or b² - 4ac.

⇒ D = (-4)² - 4(2)(3).

⇒ D = 16 - 24.

⇒ D = -8.

D < 0 = Roots are imaginary and unequal Or complex conjugate.

                                                                                                                         

MORE INFORMATION.

Conjugate roots.

(1) = If D < 0.

One roots = α + iβ.

Other roots = α - iβ.

(2) = If D > 0.

One roots = α + √β.

Other roots = α - √β.

Answer 2

Proper Question :

⠀⠀▪︎ ⠀Find the discriminant of the quadratic equation 2x² -4x + 3 =0 and the hence find the nature of its roots.

⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━⠀

⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀☆GIVEN QUADRATIC EQUATION : 2x² - 4x + 3 = 0 ,

⠀⠀⠀⠀⠀Now ,

⠀⠀By Comparing it with standard form of QUADRATIC EQUATION and it's given by :

$\qquad \dag\:\:\bigg\lgroup \sf{ Standard \:_{(QUADRATIC\:EQUATION \:)}\:\:: ax^2 + bx + c \:}\bigg\rgroup$

⠀⠀⠀⠀⠀We get ,

⠀⠀⠀⠀⠀▪︎ ⠀a = 2

⠀⠀⠀⠀⠀▪︎ ⠀b = -4

⠀⠀⠀⠀⠀▪︎ ⠀c = 3

$\dag\:\:\sf{ As,\:We\:know\:that\::}\\\\ \bigstar\:\:\bf Formula\:of\:Discriminant\::$

$\qquad \dag\:\:\bigg\lgroup \sf{ D \: = \: b^2 \:-\:4ac }\bigg\rgroup$

⠀⠀⠀⠀⠀Here , D is the Discriminant .

⠀⠀⠀⠀⠀&

⠀⠀⠀⠀⠀▪︎ ⠀If D = 0 then The roots are equal , real & rational .

⠀⠀⠀⠀⠀▪︎ ⠀If D > 0 then The roots are real , distinct & rational .

⠀⠀⠀⠀⠀▪︎ ⠀If D < 0 then The roots are imaginary & unequal.

$\qquad:\implies \sf D \: = \: b^2 \:-\:4ac \:$

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

$\qquad:\implies \sf D \: = \: b^2 \:-\:4ac \:$

$\qquad:\implies \sf D \: = \: (-4)^2 \:-\:4(2)(3) \:$

$\qquad:\implies \sf D \: = \: 16 \:-\:4(2)(3) \:$

$\qquad:\implies \sf D \: = \: 16 \:-\:8(3) \:$

$\qquad:\implies \sf D \: = \: 16 \:-\:24 \:$

$\qquad:\implies \bf D \: = \: -8 \:$

⠀⠀⠀⠀⠀Therefore,

$\qquad:\implies \bf \: -8 \< 0 \:$

$\qquad:\implies \bf D \: \< \: 0\:$

$\qquad :\implies \pmb{\underline{\purple{\: D \: \< \: 0\: }} }\:\:\bigstar$

Therefore,

⠀⠀⠀⠀⠀▪︎ ⠀The roots are imaginary & unequal.

⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━⠀

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

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

⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━⠀