Question

How do you use the discriminant to determine the nature of the roots for `-6x^2 - 12x + 90 = 0`?

Answer 1

As here, `Delta>0` and a perfect square there are two real rational roots.

Explanation:

`-6x^2 -12x +90 = 0`
The equation is of the form `color(blue)(ax^2+bx+c=0` where:
`a=-6, b=-12, c=90`

The Discriminant is given by:
`Delta=b^2-4*a*c`
`= (-12)^2-(4*(-6)*90)`
`= 144+2160 =2304`

As here, `Delta>0` and it is also a perfect square there are two real rational roots.

• Note :
  The solutions are normally found using the formula
  `x=(-b+-sqrtDelta)/(2*a)`

As `Delta = 2304`, `x = (-(-12)+-sqrt(2304))/(2*-6) = (12+-48)/-12`
`x = (12-48)/-12 = 36/12= color(green)(3`
`x = (12+48)/-12 = -60/12 =color(green)( -5`