Question

`x^2-6x+4=0`

Show that the following equation has real roots. Also find the roots:

x² − 6x + 4 = 0

Answer 1

Given: 

`x^2-6x+4=0` 

On comparing it with `ax^2+bx+c=0` 

a = 1, b = −6 and c = 4  

Discriminant D is given by: 

`D=(b^2-4ac) `

= `(-6)^2-4xx1xx4` 

= `36-16` 

= `20>0`  

Hence, the roots of the equation are real.
Roots α and β are given by: 

`α=(-b+sqrt(D))/(2c)`

`=(-(-6)+sqrt(20))/(2xx1)`

`=(6+2sqrt(5))/2`

`=(2(3+sqrt(3)))/2`

`=(3+sqrt(5))`  

`β=(-b-sqrt(D))/(2a)`

`=(-(-6)-sqrt(20))/2`

`=(6-2sqrt(5))/2`

`=(2(3-sqrt(5)))/2`

`=(3-sqrt(5))` 

Thus, the roots of the equation are `(3+2sqrt(5)) and (3-2sqrt(5))`