Final answer:
The nature of the solutions to the equation 4x²-8x+9=0 can be determined by the discriminant, which is negative. Hence, the equation has no real solutions and instead has complex or imaginary solutions.
Step-by-step explanation:
To describe the nature of the solutions to the quadratic equation 4x²-8x+9=0, we can use the discriminant. The discriminant is found by the formula b²-4ac, where a, b, and c are the coefficients of the terms in the quadratic equation ax²+bx+c=0. In this case, a=4, b=-8, and c=9.
Let's calculate the discriminant:
- b²-4ac = (-8)² - 4(4)(9)
- b²-4ac = 64 - 144
- b²-4ac = -80
Since the discriminant is negative (-80), this tells us that there are no real solutions to the equation and the solutions are instead complex or imaginary numbers.
The quadratic formula, which is x = (-b ± √(b²-4ac)) / (2a), would yield complex solutions in this case because the square root of a negative number is imaginary.