Question

Without factorization, find the nature of the roots of the quadratic equation. .
4x2- 12x + 9 = 0.​

Answer 1

Step-by-step explanation:

Here a=4,b=-12 and c=-9

D=

b

2

−4ac

D=

(−12)

2

−4(4)(−9)

D=144+144=288

Answer 2

⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀☆GIVEN QUADRATIC EQUATION : 4x² - 12x + 9 = 0 ,

⠀⠀⠀⠀⠀Now , We know that for finding nature of roots we find Discriminant ( D ) and it's given by :

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

⠀⠀⠀⠀⠀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.

⠀⠀⠀⠀⠀In the given Quadratic equation : 4x² - 12x + 9 = 0 { a = 4 , b = -12 & c = 9 }

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

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

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

$\qquad \dashrightarrow \sf D \: = \: (-12)^2 \:-\:4(4)(9) \:$

$\qquad \dashrightarrow \sf D \: = \: (-12)^2 \:-\:16(9) \:$

$\qquad \dashrightarrow \sf D \: = \: (12)^2 \:-\:144 \:$

$\qquad \dashrightarrow \sf D \: = \: 144\:-\:144 \:$

$\qquad \dashrightarrow \sf D \: = \: 0 \:$

⠀⠀⠀⠀⠀Therefore,

$\qquad \dashrightarrow \sf D \: = \: 0 \:$

$\qquad \dashrightarrow \sf 0 \: = \: 0 \:$

$\qquad \dashrightarrow \sf D \: = \: 0 \:$

$\qquad \dashrightarrow \pmb{\underline{\purple{\: D \: \ = \: 0\: }} }\:\:\bigstar$

⠀⠀⠀⠀⠀⠀Hence , The roots are equal , real & rational  .

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