How many real-number solutions does the equation have?

The equation 0 = 3x^2 - 5x - 5 can be rewritten in the standard quadratic form as 3x^2 - 5x - 5 = 0.

To find the number of real-number solutions, we can use the discriminant formula:
Discriminant = b^2 - 4ac
where a = 3, b = -5, and c = -5.

Plugging in the values, we get Discriminant = (-5)^2 - 4*3*(-5) = 25 + 60 = 85

Since the discriminant is a positive number (85), the equation has two distinct real number solutions.

are you sure it's not infinitely many solutions?

I apologize for the confusion in my previous response. The equation 0 = 3x^2 - 5x - 5 is a quadratic equation, and quadratic equations typically have either 0, 1, or 2 real number solutions.

In this case, we have calculated that the discriminant is positive (85), which means there are two distinct real number solutions to this equation. So, the equation 0 = 3x^2 - 5x - 5 has exactly 2 real number solutions. Thank you for pointing that out, and I apologize for any confusion.