Final answer:
To solve the quadratic equation -3x^2-5x+8=0, we use the quadratic formula. After calculating the discriminant, which is positive, we find two different real roots for the equation: x = -8/3 and x = 1.
Step-by-step explanation:
To solve the quadratic equation -3x^2-5x+8=0, we can use the quadratic formula which is given by x = (-b ± √(b^2-4ac))/(2a), where a, b, and c are coefficients from the equation of the form ax² + bx + c = 0. In our equation, a = -3, b = -5, and c = 8.
First, we need to determine the discriminant, which is b² - 4ac. This will tell us the number and type of roots the equation has.
The discriminant is √((-5)^2 - 4(-3)(8)) = √(25 + 96) = √121 = 11. Since the discriminant is positive, this means we have two different real roots. We can now apply the values to the quadratic formula:
x = (5 ± 11) / (2 * -3)
For the positive part of the ±: x = (5 + 11) / -6 = 16 / -6 = -8/3
For the negative part of the ±: x = (5 - 11) / -6 = -6 / -6 = 1
Therefore, the quadratic equation -3x²-5x+8=0 has two real roots: x = -8/3 and x = 1.