Question

Determine whether the following equation has real roots or not. If yes, find them:

`3x^2 + 2 - sqrt5x - 5 = 0`

Answer 1

Given:

`3x^2 + 2 - sqrt5x - 5 = 0`

`=> 3x^2 - sqrt5x - 3 = 0`

ax² + bx + c = 0

Where:

a = 3

b = −5

c = −3

D = b² − 4ac

`D = (-sqrt5)^2 - 4(3)(-3)`

= 5 + 36

= 41

Discriminant D = 41 > 0

Since 41 is positive but not a perfect square, the roots are:

Real, unequal, and irrational

`x = (-b+-sqrtD)/(2a)`

`=(-(-sqrt5) +- sqrt41)/(2xx3)`

`= (sqrt5 +- sqrt41)/6`

`= (sqrt5 + sqrt41)/6` or `= (sqrt5 - sqrt41)/6`