Final answer:
The quadratic equation −x²+6x−8=0 has two real and distinct solutions because the discriminant (Δ) is positive. These solutions can be calculated using the quadratic formula.
Step-by-step explanation:
To determine the number and type of solutions to the quadratic equation −x²+6x−8=0, we can use the discriminant method. The discriminant is found using the formula b² - 4ac, where a, b, and c are coefficients from the equation in the form ax²+bx+c=0. In our case, a=-1, b=6, and c=-8.Let's calculate the discriminant:The given quadratic equation is -x² + 6x - 8 = 0. We can determine the number and type of solutions by using the discriminant, which can be found using the formula D = b² - 4ac, where a, b, and c are the coefficients of the quadratic equation.
In this case, a = -1, b = 6, and c = -8.By substituting these values into the discriminant formula, we get D = (6)² - 4(-1)(-8) = 36 - 32 = 4.If the discriminant is positive (D > 0), then there are two distinct real solutions. If the discriminant is zero (D = 0), then there is one real solution. If the discriminant is negative (D < 0), then there are no real solutions.Since the discriminant in this case is positive (D = 4), we conclude that the quadratic equation has two distinct real solutions.Since the discriminant is positive (Δ > 0), the quadratic equation has two real and distinct solutions. These solutions can be found using the quadratic formula, x = (-b ± √Δ) / (2a).