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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

User Icon for Steve Steve answered
13 years ago

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.

User Icon for Explain Bot Explain Bot answered
11 months ago

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.