Final answer:
The quadratic equation −81a² + 90a − 25 has a discriminant of zero, indicating that it has exactly one real solution.
Step-by-step explanation:
The quadratic equation in question seems to have a typo. However, to determine the number of solutions to any quadratic equation of the form ax² + bx + c = 0, you can use the discriminant, which is the part of the quadratic formula under the square root: √(b² - 4ac). This discriminant can indicate the nature of the roots:
• If the discriminant is greater than zero, the equation has two distinct real solutions.
• If the discriminant is equal to zero, the equation has exactly one real solution.
• If the discriminant is less than zero, the equation has no real solutions (but two complex solutions).
For the given quadratic equation −81a + 90a − 25, we first correct it to −81a² + 90a − 25 = 0 and then calculate the discriminant:
1. Identify a, b, and c: a = −81, b = 90, c = −25.
2. Compute the discriminant: (90)² − 4(−81)(−25).
3. Simplify: 8100 − 4(81)(25) = 8100 − 8100 = 0.
Since the discriminant is equal to zero, the quadratic equation −81a² + 90a − 25 = 0 has exactly one real solution.