use the discriminant to determine whether the following equations have solutions that are: tow different rational solutions; two different irrational solutions; exactly one rational solution; or two different imaginary solutions. 25x^2=10x-1

25x^2 - 10x + 1 = 0

discriminant = b^2 - 4ac = 100-100 = 0

so, the roots are the same.

(5x-1)(5x-1)

Why does the discriminant determine how many roots? Well, the roots are basically -b/2a +/- the discriminant. So, if the discriminant is zero, both roots are -b/2a

If the discriminant is positive, both roots are the same distance away from -b/2a

If the discriminant is negative, there is no real square root, and we have complex roots.

To determine the types of solutions of the equation 25x^2=10x-1 using the discriminant, we first need to rewrite the equation in the form ax^2 + bx + c = 0. In this case, a = 25, b = -10, and c = 1.

The discriminant (denoted by Δ) is calculated using the formula Δ = b^2 - 4ac. Let's substitute the values into the formula:

Δ = (-10)^2 - 4(25)(1)
= 100 - 100
= 0

Now that we've found the discriminant, we can determine the types of solutions based on its value:

1. If Δ > 0, there are two different rational solutions.
2. If Δ < 0, there are two different imaginary solutions.
3. If Δ = 0, there is exactly one rational solution.

In this case, Δ = 0, which means there is exactly one rational solution.

Therefore, the given equation, 25x^2 = 10x - 1, has exactly one rational solution.