Question

Find the roots of
$4x {}^{3} + 20x {}^{2} - 23x + 6 = 0$ if two of its roots are equal.​

Answer 1

Answer:

Roots are

1/2 , 1/2 , -6

Step-by-step explanation:

Find the roots of

4x³+ 20x² - 23x + 6 = 0 if two of its roots are equal.​

Two roots are Equal

so let say

f(x) = (x-a)(x-a)(cx-b)

f(x) = (x² - 2ax + a²)(cx - b)

f(x) = cx³ - (2ac + b)x²  + (2ab + a²c)x  - a²b

Equating with

4x³+ 20x² - 23x + 6

=> c  = 4

2ac + b = - 20

=> 8a + b = -20

2ab + a²c  = -23

=> 2ab + 4a² = -23

=> 2a(-20 - 8a) + 4a² = -23

=> -40a -16a² + 4a² = -23

=> 12a² + 40a - 23 = 0

=> 12a² -6a + 46a - 23 = 0

=> 6a(2a - 1) + 23(2a - 1) = 0

=> (6a + 23)(2a - 1) = 0

=> a = -23/6  , a = 1/2

=> b =  32/3  , b = -24

a²b = -6  hence a = 1/2 & b = -24 satisfies it

(x - 1/2)²(4x + 24)

= (2x - 1)²(x + 6)

=> x = 1/2  & x = -1/6