Question

How many solutions are there to the equation 0 = 3x^2 - 10x - 5?

Answer 1

compare this equation with a`x^2` +bx+c=0
a= 3, b=-10, c=-5
Discriminant= `B^2` -4ac
= `(-10)^2` - 4*3(-5)
= 100+60
=160 which is more than 0
So it has two solutions.

Answer 2

Two

Explanation:

For a quadratic in the general form:
`color(white)("XXX")ax^2+bx+c=0`
the discriminant
`color(white)("XXX")Delta = b^2-4ac`
indicates the number of solutions:
`Delta { (< 0, "no solutions"), (= 0, "exactly one solution"),(> 0, "two solutions") :}`

In this case `a=3`, `b=-10`, and `c=-5`
so `Delta = (-10)^2-4(3)(-5) = 160 > 0`
`rArr` two solutions.

Further, the discriminant is part of the quadratic formula that gives the actual solutions:
`color(white)("XXX")x = (-b+-sqrt(Delta))/(2a)`

In this case
`color(white)("XXX")x= (5+-2sqrt(10))/2`