Question

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

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

Answer 1

Given:

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

This is a quadratic of the form:

ax² + bx + c = 0 

Where:

a = 3

b = `3sqrt5`

c = −5

D = b² − 4ac

D = `(3sqrt5​)2 − 4(3) (−5)`

= 9 × 5 + 60 

= 45 + 60

= 105

D = 105 > 0, so the equation has two real and unequal roots

Since 105 is not a perfect square, the roots are irrational

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

`= (-3sqrt5 +- sqrt105)/(2xx3)`

`= (-3sqrt5 +- sqrt105)/6`

The equation has real, unequal, and irrational roots

`= (-3sqrt5 + sqrt105)/6` or `= (-3sqrt5 - sqrt105)/6`